ترجیحات اجتماعی یا نگرانی شغلی شخصی؟ شواهد درست در روابط متقابل مثبت و منفی در محل کار

Abstract This paper provides non-experimental field evidence on positive and negative worker reciprocity. We analyze the performance reactions of professional workers to fair and unfair wage allocations in their natural environment. The objects of interest are professional soccer players in the German Bundesliga. This environment enables us to circumvent the main problems of observational studies on reciprocity because there is substantial transparency in individual player values and performance. Our main finding is that workers exhibit both positive and negative reciprocity toward employers who deviate from a player’s perception of a fair market wage. This perception of a fair wage follows from a Mincer-type wage equation that incorporates a worker’s past performance. The different results between changing and non-changing players are in line with theories of fairness perception but cannot be explained by private information from the employers or the personal career concerns of the players. Altogether, our findings provide strong evidence for the external validity of previous laboratory results on gift exchange in the labor market.

Introduction Consider the following scenario. Tom just finished his B.A. in Management and applies for a job in the retailing industry. From a business magazine, he knows the average salary of new job entrants in this industry. After a series of job interviews, Tom receives only one job offer, which offers a salary that is considerably lower than the average salary level. Because he does not want to be unemployed, Tom accepts the offer and begins working in the industry. Now imagine a situation in which the same average wage level for new entrants applies but in which Tom’s wage offer substantially exceeds the average salary. Again, he accepts the offer and starts working in the industry. Will it matter for Tom’s subsequent working performance which of the two scenarios actually occurs? And to what extent will his behavior reflect fairness considerations? This paper aims to answer these questions and reports evidence from a non-experimental field study that explores whether paying above-market wages induces workers to improve their performance. In addition, we test whether paying below-market wages induces workers to reduce their performance. The literature on gift exchange in the labor market (Akerlof, 1982, Akerlof and Yellen, 1988 and Akerlof and Yellen, 1990) assumes that Tom uses the average salaries of similar others (i.e., newcomers to the industry) to form his reference wage, against which he compares his actual salary. If Tom perceives himself to be underpaid, he will reduce his performance, and if he perceives himself to be overpaid, he will increase his performance. Overall, this literature proposes that workers and employers engage in reciprocal gift giving, where the size of the gift from the worker is his performance in excess of the minimum work standard while the size of the gift from the firm is represented by wages in excess of what the worker could earn at another firm. This gift-exchange view of labor relations is supported by findings from numerous laboratory experiments (Pereira, Silva, & e Silva, 2006; see Fehr and Gächter, 1998 and Fehr and Gächter, 2000 for overviews). The inclination of individuals to reward those who have been kind to them and to punish others who have been unkind to them has been labeled positive and negative reciprocity, respectively (see the discussion in Rabin, 1993). Studies in support of positive and negative reciprocity among individuals are now so numerous that several formal theories of reciprocal behavior have been developed (Cox, Friedman, & Gjerstad, 2007; Dufwenberg & Kirchsteiger, 2004; Falk & Fischbacher, 2006). However, the observation that these theories are mainly built on laboratory evidence has recently become a concern for several researchers (e.g., Levitt & List, 2007), and the extent to which this evidence translates into the field remains the subject of ongoing discussion. Whereas several field experiments on worker reciprocity have now been performed (Bellemare and Shearer, 2009, Cohn et al., 2009, Gneezy and List, 2006 and Hennig-Schmidt et al., 2010; Kube et al., forthcoming(a)), their findings are less clear, and the effects detected are usually smaller than those detected in laboratory studies. This conflict between laboratory and field studies is sometimes considered evidence that laboratory findings do not translate into the field (Levitt & List, 2007). It may be too soon to draw such conclusions, as the number of existing field studies remains limited. This lack of studies is particularly severe in the area of observational field studies on worker reciprocity. These studies have encountered considerable problems, such as a lack of good proxies for worker effort, a mix of incentives for workers, productivity differences among workers, and the influence of workers’ strategic considerations when choosing effort levels (Falk & Heckman, 2009). Moreover, alternative wages for employees are often not observable for researchers (Bellemare & Shearer, 2009). While these problems are specific to observational studies, experimental and observational field studies share a common shortcoming: studies that jointly analyze positive and negative reciprocity are scarce, 1 which makes it difficult to compare previous reciprocity findings across different studies. In this paper, we aim to close this gap in the literature and present observational field evidence on both positive and negative worker reciprocity. We aim to circumvent the aforementioned measurement problems by using seasonal data from professional soccer players in their natural environment (the German Bundesliga) over a five-year period. The choice of this industry has considerable advantages for the purpose of our study. First, for each worker, objective performance values and employer information can be obtained for every year under study (Kahn, 2000). In particular, we use an expert evaluation of player performance to reveal productivity differences across players within the same tactical position. This effort proxy is much more reliable than those used in previous field studies (e.g., Lee & Rupp, 2007). Information on the team that a player stayed with during a particular season allows us to address concerns about personal reputation as a motivational factor in player performance. Second, although player salaries are not publicly available, there is considerable transparency in the player market (Torgler, Schaffner, Frey, & Schmidt, 2008) because the highly renowned German Kicker Sportmagazin provides market value estimates for the players in the German Bundesliga. Previous studies have most frequently used these wage data, because the data’s reliability has been judged to be high ( Franck & Nüesch, 2011, 2012; Frick, 2007, Haas et al., 2004, Kern and Süssmuth, 2005 and Torgler and Schmidt, 2007). These data allow us to compare a player’s actual wage with his fair market wage. Such entitlements have previously been documented to trigger fairness perceptions (see Fehr, Goette, & Zehnder 2009 and the references therein). Third, professional sports players do not face a complex mix of incentives. Performance has to be provided on the field and is constantly observed by managers and fans. 2 In line with the gift-exchange view, we use the average wage of similar others to form a player’s reference wage. In a first step, we obtain this reference wage as the prediction from a Mincer-type wage equation that includes a player’s observable characteristics (e.g., age, experience, tenure) and his performance in the previous season. In a second step, the player is assumed to form the fairness evaluation of his current wage by comparing it to the reference wage and, subsequently, to choose his performance level. Our empirical results support earlier experimental findings in the field in that we find a small but statistically significant effect for wages that deviate from market levels: depending on whether wages are lowered or increased relative to the reference wage, output elasticities amount to −0.25 and 0.10, respectively. Whereas performance reductions in response to underpayment are in line with negative reciprocity, they are never optimal for purely self-interested players (because future wages are increasing in performance). In contrast, performance increases in response to overpayment may well reflect the career concerns of personally self-interested players. To test whether the positive effect should instead be attributed to fairness concerns, we compare performance responses across changing and non-changing workers. Whereas both groups face comparable career incentives, findings by Gneezy and List (2006) reveal that positive reciprocity decreases with the duration of an ongoing working relationship. We thus predict that positive reciprocity is higher for players who have recently changed to a new team than for players who have remained with their former team. Our empirical results support this prediction. The remainder of this study is organized as follows. The next section derives our testable hypotheses. Section 3 discusses our data and presents our estimation approach to analyze fairness considerations and worker performance. Section 4 presents the empirical results, and Section 5 concludes.

. Results In this section, we first present the estimation results from our wage equation. Next, we document descriptive evidence that worker performance differs between over- and underpaid workers. To determine the extent to which this difference is driven by positive reciprocity, negative reciprocity, or a combination of both, we subsequently present results from a linear regression approach. Finally, we aim to shed light on the motivational concerns of workers that underlie these results. 4.1. Predicting fair wages Table 3 displays the estimation results of a linear regression model for a player’s next-season wage on his performance in the previous season and on his individual characteristics. All coefficients in Table 3 reveal the expected signs. In line with our reasoning from the theoretical section, we find the previous season’s performance to improve this season’s wage level for a player. Similarly, greater experience leads to higher wages, in terms of both absolute number of match appearances and playing minutes in this season. Interestingly, professional soccer players also reveal the well-known nonlinear relation between age, experience and wages, as reflected in the obtained concavity. Analyzing these concave relationships in greater detail, we find player values to be maximized, ceteris paribus, around the age of 22 years and after 219 matches in the German Bundesliga. Comparing these values to the summary statistics in Table 1, we see that these values lie in the observed range for players in our sample. Whereas the point estimates on firm-specific human capital also reveal a concave relationship, these coefficients are not statistically significant. Table 3. Estimation results for the wage equation. Variable Coefficient Std. error Performance 1072.37⁎⁎⁎ (220.05) Playing time 445.14⁎⁎⁎ (70.88) Age 678,421.40⁎⁎ (270,986.10) Age2 −15,628.57⁎⁎⁎ (5954.92) Experience 16,940.47⁎⁎ (6928.95) Experience2 −38.65⁎⁎⁎ (13.41) Team tenure 17,837.71 (46,137.98) Team tenure2 −485.61 (4505.67) Player fixed effects Yes Season fixed effects Yes New team fixed effects Yes Position fixed effects Yes R2 (within) 0.41 N 1009 Note : Displayed are the OLS estimation results for the estimation equation View the MathML sourcewijtp=β0+β1performancei,t-1+β2minutesi,t-1+β3ageit+β4ageit2+β5experi,t-1+β6experi,t-12+β7tenureit+β8tenureit2+αiγt+θj+πp+εijtp, where wijtp denotes player i’s wage in season t, performancei,t−1 denotes seasonal performance in season t − 1, and minutes t−1 denotes player i’s total minutes on the playing field in season t − 1. The included observations are all players in the sample that appeared at least 90 min on the playing field for a given season. Heteroskedasticity-robust standard errors are given in parantheses. * Statistical significance at the 10% level. ⁎⁎⁎ Statistical significance at the 1% level. ⁎⁎ Statistical significance at the 5% level. Table options 4.2. Subsequent player performance Building on the average market wage to serve as a reference wage for fairness considerations, we are able to derive for every player in each season, whether he was in a “fair” state, where his wage exceeded his reference wage, or whether he was in an “unfair” state. A simple descriptive analysis reveals the performance levels of the workers to differ substantially across both states. Whereas the average subsequent performance value of players who earn at least their reference wage equals 812.28, the average performance of players who are in an unfair state equals 707.21. This difference is highly statistically significant (a Wilcoxon–Mann–Whitney test 4 yields z = 7.40, p-value < 0.001). This result is in line with Hypothesis 1, which states that wage deviations from the reference wage lead to performance changes. Although this result is already interesting in itself, the adoption of a linear regression model is needed to determine whether this finding should be attributed to better performance from overpaid workers (Hypothesis 1a), worse performance from underpaid workers (Hypothesis 1b), or a combination of the two. Column 1 in Table 4 shows the estimation results for Eq. (3).5 The associated performance sensitivities to the over- and underpayment amount are 0.100 and −0.249, respectively. In addition, we document a statistically significant negative influence from higher absolute wages on worker performance. Although we were surprised by this coefficient, this finding could result from the downward stickiness of player wages. If wages are slightly increasing with age, but performance is decreasing with age, downward sticky wages may lead to the observed effect. In columns 2 and 3, our model is subsequently extended to control for team and player position effects, respectively. Table 4. Estimation results on player performance (all players). Variable Model 1 Model 2 Model 3 Coefficient Std. error Coefficient Std. error Coefficient Std. error Log(Wage) −0.168⁎⁎⁎ (0.054) −0.172⁎⁎⁎ (0.047) −0.172⁎⁎⁎ (0.048) Fair 0.100⁎⁎ (0.045) 0.082⁎⁎ (0.042) 0.083⁎⁎ (0.041) Unfair −0.249⁎⁎⁎ (0.085) −0.261⁎⁎⁎ (0.081) −0.259⁎⁎⁎ (0.083) Player fixed effects Yes Yes Yes Season fixed effects Yes Yes Yes New team fixed effects No Yes Yes Position fixed effects No No Yes Lambda −2.49 −3.18 −3.12 R2 0.04 0.21 0.21 N 975 975 975 Note: Displayed are the OLS estimation results for the estimation equation log(performanceijtp) = β0 + β1 log(wageijtp) + β2Fairijtp + β3Unfairijtp + αi + γt + θj + πp + εijtp , where log(performanceijtp)log(performanceijtp) denotes player i’s log performance in season t, log(wijtp) denotes his logarithmic wage in season t, and Fair and Unfair denotes player i’s extent of over- or underpayment, respectively. Heteroskedasticity-robust standard errors that have been adjusted for clustering on the player level are given in parantheses. In line with our theoretical propositions, the documented statistical significance on Fair and Unfair is based on one-sided tests (in none of the models, however, does the level of statistical significance change when two-sided tests are applied). * Statistical significance at the 10% level. ⁎⁎⁎ Statistical significance at the 1% level. ⁎⁎ Statistical significance at the 5% level. Table options We find workers’ responses to fair and unfair wage allocations to be extremely robust to the inclusion of team and position fixed effects: independent of these additional controls, the performance sensitivity to overpayment remains at approximately 0.08 whereas the absolute sensitivity to underpayment remains at approximately 0.26. The relative size of positive to negative sensitivities is −3.15, which suggests workers’ disposition to punish underpayment to be stronger than the disposition to reward overpayment. A formal test, however, fails to reject the null that the ratio of positive to negative reciprocal behavior (lambda) is significantly different from −1. Therefore, we find only illustrative evidence for Hypothesis 2. Nevertheless, in line with Hypotheses 1a and 1b, we find both positive and negative deviations from the reference wage to influence worker performance. 4.3. Motivational concerns for reciprocity Are the documented findings in Table 4 attributable to worker reciprocity? Or do these findings mirror personal self-interest in receiving future gifts from employers? For underpaid workers, the answer to the influential role of reciprocity is affirmative. The reader will recall from Table 3 that seasonal performance is positively associated with subsequent wage expectations. This association implies that workers’ reaction of reducing their performance in response to unfair payments is not in their best self-interest. Homo oeconomicus would therefore never engage in such kind of behavior. However, this finding is in line with our theoretical predictions for homo reciprocans. To determine whether positive reciprocity is the mechanism underlying performance increases for fairly treated workers, we separate players who changed teams between season t − 1 and season t from those players who remained with their team. Whereas these two groups of players face comparable career incentives (or other aspects of personal self-interest), previous findings by Gneezy and List (2006) reveal that positive reciprocity decreases with the duration of an ongoing working relationship. In line with these findings, we predict that positive reciprocity is higher for players who have moved to a new team (i.e., who are at the beginning of a new working relationship) than for players who have already been with their team for some time. Table 5 presents separate estimation results for changing and non-changing players. The results show the previous pattern of positive and negative performance effects from wage differentials for non-changing workers. In line with our expectation, changing players are much more motivated to repay gifts (high wages) from their new employers; in comparison to non-changing players, changing workers’ performance sensitivity to overpayment is approximately 44 times greater. Table 5 also shows that changing players do not punish “unfair” wages. This observation is in line with the gift-exchange view. Recall that the size of the gift from the employer is equal to the wages in excess of what a player could earn with another team. If a player is moving to a team that pays him less than his reference wage, then all other wage offers must have been below the wage offer that he eventually accepted (we consider the importance of non-monetary utility in the decision making of changing players below); despite its low level, the current wage actually represents a gift from the new employer. In consequence, the player has no reason to punish the new team for being “unfair”. Table 5. Estimation results on player performance (changing vs. non-changing players). Variable Non-changing players Changing players Coefficient Coefficient Coefficient Coefficient Logwage −0.121⁎⁎ −0.120⁎⁎ 0.068 0.470 (0.054) (0.055) (0.369) (0.553) Fair 0.067⁎ 0.069⁎ 3.035⁎⁎⁎ 3.553⁎⁎⁎ (0.048) (0.048) (0.937) (1.091) Unfair −0.254⁎⁎⁎ −0.252⁎⁎⁎ 0.383 0.645 (0.096) (0.098) (0.412) (0.707) Player fixed effects Yes Yes Yes Yes Season fixed effects Yes Yes Yes Yes Position fixed effects No Yes No Yes Lambda −3.79 −3.65 0.13 0.18 N 846 846 129 129 R2 0.05 0.06 0.69 0.74 Note : Displayed are the OLS estimation results for the estimation equation log(performanceijtp)=β0+β1log(wijtp)+β2Fairijtp+β3Unfairijtp+αi+γt+θj+πp+εijtplog(performanceijtp)=β0+β1log(wijtp)+β2Fairijtp+β3Unfairijtp+αi+γt+θj+πp+εijtp, where log(performanceijtp)log(performanceijtp) denotes player i’s log performance in season t, log(wijtp) denotes his logarithmic wage in season t, and Fair and Unfair denotes player i’s extent of over- or underpayment, respectively. Heteroskedasticity-robust standard errors that have been adjusted for clustering on the player level are given in parantheses. In line with Table 4, the documented statistical significance on Fair and Unfair is based on one-sided tests. ⁎⁎⁎ Statistical significance at the 1% level. ⁎⁎ Statistical significance at the 5% level. ⁎ Statistical significance at the 10% level. Table options We would also like to note that Table 5 supports our previous interpretation that downward wage stickiness drives the negative coefficient on log(wage). Here, we observe that the negative effect is nonexistent for those players who change teams; in these situations, a new contract will be much more flexible in the downward direction than a prolongation of an existing contract for non-changers. The detected difference raises the question of whether the distribution of unfair and fair wage deviations, wages, and performance levels differs between the two groups. We address this concern in Table 6 and provide Wilcoxon–Mann–Whitney tests for key worker characteristics for changing and non-changing players. Table 6. A comparison of player characteristics across changing-and non-changing players. Variable Non-changing players Changing players z-Statistic Mean Std. Dev. Mean Std. Dev. Log (Wage) 14.35 0.70 14.34 0.68 0.43 Fair 0.22 0.44 0.15 0.30 1.46 Unfair 0.33 0.41 0.33 0.38 −0.67 Performance 752.51 226.54 732.74 251.70 0.47 Time on pitch 1593.39 868.36 1538.60 849.76 0.69 Age 27.09 3.78 26.57 3.27 1.47 Experience 92.76 76.77 90.05 61.51 −0.78 N = 846 N = 129 Note: Displayed are test-statistics for Wilcoxon–Mann–Whitney tests between changing and non-changing players. ***, **,* denote statistical significance at the 1%, 5%, and 10% level, respectively. Table options Table 6 does not reveal any meaningful statistical differences for the explanatory variables from our performance or wage equation, i.e., for wage, Fair, Unfair, age, playing minutes, and performance levels, the groups of non-changing and changing workers are comparable. Therefore, our finding cannot be attributed to differences across members the two groups. However, a potential concern about our descriptive analysis is that we do not incorporate non-monetary utility sources for players in Table 6. Specifically, rational players should change teams whenever doing so increases their expected career earnings. Therefore, players might be willing to trade immediate short-term wages with non-monetary aspects, such as expected playing time with the new team, the opportunity to become a team leader, or to play in international competitions, all of which could ultimately increase the player’s wages at some later point. If this is the case, we will have another, non-fairness-related explanation for why changing players do not punish low wages. Although we are unable to measure precisely a player’s playing time expectations (which might partly relate to unobservable contractual arrangements) and opportunity to become a team leader, we are able to objectively measure each team’s participation in international, European competitions. Therefore, we chose to integrate only this non-monetary aspect in the descriptive analysis. Specifically, we collected information on all Champions League (CL) and European League (EL) participants from the German Bundesliga for the 2001/02–2005/06 period and created a dummy variable, international competitionjt, which takes the value 1 if team j played in the CL or EL in season t and takes the value 0 otherwise. We then calculated for each player i the difference in international competitionjt between his team j in season t − 1 and his team j’ in season t (for non-changing players j equals j′). Note that this difference variable, denoted by d(international competitionijt), can take three values: (1) d(international competitionijt) = −1, player i’s current team does not play in an international competition but his previous team did. (2) d(international competitionijt) = 1, player i’s current team does play in an international competition but his previous team did not play in an international competition in season t − 1. (3) d(international competitionijt) = 0, no difference in the international competition status between player i’s team in season t and his previous team in season t − 1. If players are willing to forego higher wages for the previously unavailable opportunity to play in an international competition, we predict that (a) on average, d(international competitionijt) is higher among the group of changers than among the group of non-changers, and (b) changing players are more likely to be in the unfair state when the value of the difference variable is greater. Therefore, in our model, unfairijt and d(international competitionijt) are positively correlated. However, none of these predictions is supported by the data: in contrast to (a), we observe that the value on d(international competitionijt) is in fact lower among changing players (−0.028) than among non-changing players (0.073). Although this difference is statistically significant at the 5% level (Wilcoxon–Mann–Whitney test: z = 2.05), this effect does not suggest that players change teams often for the opportunity to play in international competitions (relative to their experience with their previous employer in the last season). Analyzing the correlation between unfair and d(international competition) for changing players, we observe a value of −0.0668, which is not statistically significant at any conventional level. Although this evidence is not conclusive, we do not believe that the difference in the performance for changing and non-changing players mainly reflects the willingness of changing players to trade off short-term wages against non-monetary utility components that might improve the player’s career opportunities in the long run. In consequence, the detected performance differences between changing and non-changing workers rule out another easy alternative explanation for our results, namely, that the detected behavior stems from omitted variable bias. A critical reader might argue that the documented pattern is most easily explained by omitted variables in our wage prediction because players perform better when their wage is higher than the amount predicted from Eq. (1) but perform worse when their wage is lower than predicted. However, omitted variable bias cannot explain the asymmetry in behavior from changing workers—why should they not perform worse when their wage is lower than predicted but much better when their wage is higher than predicted? However, from the perspective of fairness perception, this asymmetry is easily explained. Nevertheless, we decided to address the omitted variable bias problem from yet another perspective in the next subsection. 4.4. Robustness checks The above results reveal that players’ performance response to fair and unfair wage allocations follow a clear pattern that is perfectly in line with the gift-exchange view and previous laboratory evidence. We now show that our findings are robust to the inclusion of a player’s current wage in the wage equation (Section 4.4.1), or performance expectations, and injury proxies (Section 4.4.2). 4.4.1. Using current wages to predict the reference wage Including a player’s current wage as an additional explanatory variable in Eq. (1) may be based on two theoretical grounds. First, the reader should recall from Section 2 that a worker’s current wage is traditionally considered the relevant reference wage for fairness evaluations of incumbent workers. Although we emphasized the importance of market values in the highly transparent soccer industry, fairness perceptions may still be conditioned on current wage levels. Including current wages in Eq. (1) thus provides a more general approach with respect to formation of the reference wage. Second, the incorporation of a player’s current wage level reduces any persistent omitted variable bias in our wage equation. That is, if there are any long-term factors missing from Eq. (1) that determine the player’s value for a team, the inclusion of the current wage should help to mitigate this problem. Table 7 displays estimation results for our performance equation when a player’s current wage is included to form his fair wage perception. The results show a qualitatively identical pattern for players’ response to fair and unfair wage levels and support our previous conclusions.6 Table 7. Robustness check 1: estimation results on player performance (all players). Variable Model 1 Model 2 Model 3 Coefficient Std. error Coefficient Std. error Coefficient Std. error Log(Wage) −0.205⁎⁎⁎ (0.064) −0.204⁎⁎⁎ (0.053) −0.199⁎⁎⁎ (0.053) Fair 0.180⁎⁎⁎ (0.043) 0.158⁎⁎⁎ (0.034) 0.159⁎⁎⁎ (0.037) Unfair −0.300⁎⁎⁎ (0.092) −0.320⁎⁎⁎ (0.084) −0.314⁎⁎⁎ (0.085) Player fixed effects Yes Yes Yes Season fixed effects Yes Yes Yes New team fixed effects No Yes Yes Position fixed effects No No Yes Lambda −1.67 −2.03 −1.97 R2 0.06 0.21 0.21 N 909 909 909 Note: Displayed are the OLS estimation results for the estimation equation log(performanceijtp) = β0 + β1 log(wageijtp) + β2Fairijtp + β3Unfairijtp + αi + γt + θj + πp + εijtp , where log(performanceijtp)log(performanceijtp) denotes player i’s log performance in season t, log(wijtp) denotes his logarithmic wage in season t, and Fair and Unfair denotes player i’s extent of over- or underpayment, respectively. For the fair wage prediction, we included a player’s current wage in addition to the explanatory variables in Eq. (1). Heteroskedasticity-robust standard errors that have been adjusted for clustering on the player level are given in parantheses. In line with our theoretical propositions, the documented statistical significance on Fair and Unfair is based on one-sided tests (in none of the models, however, does the level of statistical significance change when two-sided tests are applied). ** Statistical significance at the 5% level. *Statistical significance at the 10% level. ⁎⁎⁎ Statistical significance at the 1% level. Table options 4.4.2. Using performance expectations to predict the reference wage At the time of contract negotiations, players have certain expectations about their future performance. As these expectations are missing from our wage regression (1), some readers might be concerned that our previous findings merely reflect alternating player performances in the presence of roughly constant player wages. Specifically, consider a player who was injured for the majority of season t − 1 but recovered before the beginning of season t. This player should be expected to perform better in season t than in season t − 1, suggesting that our previous model underestimates his true reference wage for season t (because of his low values for playing minutes and performance). If, at the same time, his true wage did not change significantly, he would be likely to end up in a fair state. In combination with improved performance after his recovery, this lack of a change might give rise to the previously detected pattern. Unfortunately, we were unable to incorporate a player’s exact injury status in our wage regression because this information is not publicly available for our sample period. Therefore, we must construct a proxy variable. We start by calculating for each player the ratio of his playing time in season t − 1 to his playing time in season t − 2. The idea is that if a player had been severely injured in season t − 1, then his playing time in this season should have been considerably lower than his playing time in season t − 2. We then create a dummy variable, injury, which takes the value 1 if a player’s playing time ratio is among the lowest decile of all players in our sample. Note that if this variable correctly picks up on a player’s intermediate injury status in season t − 1, we expect that this variable will have a positive effect on a player’s reference wage, as his expected performance in season t is likely to be underestimated from his performance in season t − 1 (when he was injured). In addition to this injury status variable, we also adjust our wage regression model (1) by including a player’s performance expectation based on observables. To this end, we first run a linear regression model of a player’s performance in season t on his playing minutes in season t − 1, his age in season t, his experience at the end of season t − 1, and his tenure with the team in season t. For age, experience, and tenure, we also incorporate squared terms in the regression to capture nonlinearities between performance and these regressors. Therefore, the fitted values of this regression capture the expected performance of a player given his observable characteristics at the end of season t − 1 (such as his experience and previous playing time) and deterministic information in season t (such as his age or tenure). The associated estimates for the wage equation (given in Table A.1 in the Appendix) show a positive yet nonsignificant effect of the performance expectation variable. In line with our expectations, the injury status variable positively affects a player’s expected wage. The estimated coefficient is very large (385,226.10) and statistically significant at the 5% level. We note that most of the other variables maintain their statistical significance, even in the presence of the performance expectation term. Table 8 displays the estimation results for our performance equation when we control for injury status and performance expectations based on observables. Except for the statistical nonsignificance of fair in Model 1, the results show a qualitatively identical pattern in players’ responses to fair and unfair wage levels and corroborate our previous findings. Table 8. Robustness check 2: estimation results on player performance (all players). Variable Model 1 Model 2 Model 3 Coefficient Std. error Coefficient Std. error Coefficient Std. error Log(Wage) −0.160⁎⁎⁎ (0.057) −0.174⁎⁎⁎ (0.055) −0.174⁎⁎⁎ (0.055) Fair 0.064 (0.053) 0.074⁎⁎ (0.033) 0.074⁎⁎ (0.033) Unfair −0.255⁎⁎⁎ (0.097) −0.281⁎⁎⁎ (0.096) −0.280⁎⁎⁎ (0.098) Player fixed effects Yes Yes Yes Season fixed effects Yes Yes Yes New team fixed effects No Yes Yes Position fixed effects No No Yes Lambda −3.98 −3.80 −3.78 R2 0.04 0.21 0.21 N 982 982 982 Note: Displayed are the OLS estimation results for the estimation equation log(performanceijtp) = β0 + β1 log(wageijtp) + β2Fairijtp + β3Unfairijtp + αi + γt + θj + πp + εijtp , where log(performanceijtp)log(performanceijtp) denotes player i’s log performance in season t, log(wijtp) denotes his logarithmic wage in season t, and Fair and Unfair denotes player i’s extent of over- or underpayment, respectively. For the fair wage prediction, we included a player’s injury status in season t − 1 and performance expectations for season t in addition to the explanatory variables in Eq. (1). Heteroskedasticity-robust standard errors that have been adjusted for clustering on the player level are given in parantheses. In line with our theoretical propositions, the documented statistical significance on Fair and Unfair is based on one-sided tests (in none of the models, however, does the level of statistical significance change when two-sided tests are applied. * Statistical significance at the 10% level. ⁎⁎⁎ Statistical significance at the 1% level. ⁎⁎ Statistical significance at the 5% level. Table options Concluding our section on the empirical results, we document a positive correlation between fair wages and team success. Whereas the average subsequent team ranking for players with an unfair wage allocation is 9.84, it is substantially better, amounting to 7.18, for teams with fairly paid players (Wilcoxon–Mann–Whitney test: z = −7.90; p < 0.01). These results are particularly interesting, as they suggest the theoretically predicted positive relationship between fair treatments of workers and firm success.