Shirking on the court: testing for the incentive effects of guaranteed pay.

Berri, David J. ; Krautmann, Anthony C.

INTRODUCTION

Opportunistic behavior can arise in the employment relationship between a principal (the employer) and agent (the employee) when the labor contract is incomplete or when the principal is unable to distinguish shirking behavior from below-average realizations of a stochastic process. In such cases, economists expect the evolution of self-enforcing labor contracts that contain incentive-compatible stipulations to induce workers to put forth their maximum efforts. In general, such mechanisms impose losses on cheaters greater than any gains to be received from shirking (e.g., threat of termination, efficiency wages). This agency problem has a long history in the literature (Alchian and Demsetz 1972; Holmstrom 1979), and has been applied to a number of different labor outcomes, including union seniority rights, the academic institution of tenure, and long-term contracts of professional athletes.

With respect to the contracts of professional athletes, the principal-agents problem has been extensively investigated. Such interest is not surprising given the publicity surrounding the large long-term contracts given to superstars like Kevin Garnett and Alex Rodriguez. The common perception is that players become lazy and expend less effort once they have signed a long-term contract. This sentiment was expressed by Dan O'Brian, former vice president of negotiations for the Cleveland Indians Baseball: "The experience of individual clubs, and the industry as a whole, is that for whatever reason, the player's performance is not the same following the signing of a new multiyear contract" (Sporting News 1986). If true, then why do owners continue to offer long-term contracts?

Labor theory suggests that the employment contract should evolve to contain an array of mechanisms aimed at deterring opportunistic behavior (Maxcy et al. 2002). Such an evolution is made possible by a number of important characteristics of this labor market. For one, a player's performance is easily observed and scrutinized, hence his behavior is constantly being monitored by coaches, managers, owners, and fans (Fort 2003). Second, many contracts now include incentive clauses that tie individual and team performances to compensation. Finally, there is the possibility that a shirker will develop a bad reputation, reducing his likelihood of securing a lucrative long-term contract allocated to only the very elite athletes. (1)

As noted in the literature, even if a player wished to reduce effort, shirking within the course of a game may be difficult. For a player to act opportunistically, it is necessary that he have substantial control of his effort. That is, although it is certainly possible that a player can consciously decide whether to run out a pop-up or dive for a sinking ball in the outfield, one must ask whether the player has the same degree of control over his effort when it comes to how hard he tries to hit a curveball, throw a slider, or make a fade-away jump shot.

One possibility, though, is that on-field performance is a function of a player's effort both inside and outside the game. That is, effort must be exerted in practice, in the weight room, in the off-season, and with respect to diet. A player may completely exert himself in the game, but if his effort to keep in shape is lacking, then his performance during the season can be impaired. Consider the case of Vin Baker, a player who appeared in four NBA All-Star games from 1995 to 1998, was named to the All-NBA second team in 1998, and won an Olympic gold medal in 2000. After signing an $87 million contract in 1999, Baker's per game scoring went from 16.6 points in for the 1999-2000 campaign to 12.2 in 2000-2001. For the 2002-2003 campaign he averaged only 5.2 points per contest. What explains this rapid decline in performance? According to Paul Westphal (Baker's head coach from 1999 to 2001), Baker's decline was caused by a "lack of professionalism off the court" (quoted in O'Neil 2003).

The empirical evidence regarding strategic behavior of professional athletes is mixed, with perhaps the strongest evidence suggesting that the willingness to play with injuries is adversely affected by job security. Lehn (1982) was one of the first to look at the effect of long-term labor contracts on player durability. Using a sample of the earliest cohort of Major League Baseball (MLB) free agents, Lehn found evidence of opportunistic behavior in that players with long-term contracts were more likely to end up on the disabled list the following season. Using a more recent sample of free agents, however, Maxcy et al. (2002) found no evidence of shirking on a player's subsequent durability. The empirical evidence on whether on-field performance is affected by proximity to signing a long-term contract is also mixed. Although some studies have found evidence of shirking after signing a long-term contract (Marburger 2003; Scroggins 1993), others have not (Krautmann 1990; Maxcy 1997; Maxcy et al. 2002).

Much of the research on opportunistic behavior has focused on MLB, even though long-term contracts are found throughout the sports industry. For example, according to USA Today, 70% of the players in the National Basketball Association (NBA) during the 2002-2003 season played under a contract that was three years or longer in length; over 14% had a contract that was at least seven years long. (2) Consequently, if long-term contracts do have an adverse effect on effort, one might expect to find convincing evidence in the labor market in the NBA.

We find that the evidence of shirking to be mixed. Depending on how one measures an NBA player's productivity, we find evidence both for and against the shirking hypothesis. When using a measure of productivity employed by both the industry and the media, we find weak evidence of a falloff in performance after signing a long-term contract. But when a more sophisticated measure of productivity is used, one more closely aligned with an economic definition of marginal productivity, we find no support for the shirking hypothesis.

II. MODEL OF SHIRKING

In this section, we begin by constructing a model of a player's productivity in the labor market of professional basketball. Our primary objective is to test whether job security causes players to "slack off" immediately after signing long-term player contracts. In Maxcy, et al. (2002), the authors attempted to isolate shirking behavior in MLB by looking at whether a player's deviation from expected performance was affected by proximity to a free agent contract. In a similar fashion, we directly model the changes in a player's productivity ([DELTA]PROD) from year (t - 1) to year t as a function of player characteristics (PLAYER), team factors (TEAM), and measures of contractual influence (SHIRKING):

(1) [DELTA]PROD = f(PLAYER, TEAM, SHIRKING).

The first player characteristic we consider is the aging effect on a player's productivity. From prior work on the subject, we have found that a player's productivity tends to be surprisingly stable across his career, with a noticeable spike upward in his first two years and a steady decline occurring after a player's 12th year of experience. For this reason, we included a dummy variable for players with 2 years of experience (D2) and another for those who have played more than 12 seasons (D12).

Beyond the aging effect on productivity, one would expect that injuries would adversely affect performance. Without data on players' injuries, however, we are forced to use an alternative proxy. Given that injured players are typically held out of games, the number of games played should suffice as a crude measure of injuries. As such, an increase in injuries implies a decrease the number of games played ([DELTA]GP < 0), hence reducing productivity.

Beyond the player variables, three team characteristics can also potentially impact a player's level of output: managerial quality, teammate productivity, and roster turnover. The first factor listed has been the subject of a substantial amount of literature. (3) The work presented here follows from Kahn (1993), where individual performance in MLB was connected to a measure of managerial quality. Kahn measured managerial quality via the construction of a variable that linked managerial salary to both the manager's years of experience and lifetime winning percentage. Because limitations of the data prevented us from constructing a similar variable, we instead used the coach's experience (CEXP) and lifetime winning percentage (CWPCT) to control for managerial quality. Ceteris paribus we expect that a player's productivity rises with the quality of the coaching input.

Player productivity is also impacted by the actions of his fellow teammates. A common argument offered by the media is that better players increase the productivity of their teammates, even though such a view runs counter to the basic economic notion of diminishing returns. Given that shot attempts in a game are finite, if one player takes more shots, his teammates must take fewer. A similar argument extends to other facets of productivity, including rebounding and steals. To settle this debate between common perception and standard economic theory, we examine how the productivity of his teammates impacts the productivity of the player in question. This is accomplished by the change in team wins across seasons ([DELTA]TMWINS); with diminishing returns, we would expect that a player's productivity falls as he plays on a better team because his teammates are contributing a greater amount to the overall team effort. (4)

The final team characteristic to be considered is team chemistry. In general, team chemistry is an elusive concept, frequently cited but rarely defined. In this article, it will be assumed that team chemistry refers to the ability of team members to work together in a fashion that enhances each player's and hence the team's productivity. To some extent, this issue is addressed by the previously discussed issue of team productivity. Beyond this variable, the ability of teammates to work together in a positive fashion is likely to be impacted by the number of new players each team acquires in the off-season. Following the work of Berri and Jewell (2004), we examine the difference in the number of returning player minutes on a player's team ([DELTA]ROSTER). (5) A priori, more stable rosters (i.e., a decline in this factor) should lead to an increase in a player's performance.

The final factor we consider deals with shirking, the primary subject of this inquiry. One means of detecting shirking is a simple dummy variable (SIGNED) for players in their first year after signing a long-term contract (defined as greater than two years in length). (6) If shirking behavior is common, then we would find a negative effect of SIGNED on productivity. Beyond a simple dummy variable, we also wonder if the length of a contract impacts player productivity. Hence, an alternative means of detecting shirking behavior is to interact SIGNED with the length of the contract (SIGNED x LENGTH). Again, if opportunism is prevalent, then we would find this interaction term to be negatively related with productivity. Finally, with annual salaries of professional basketball players averaging in the millions, it is possible that what appears to be shirking behavior is the result of some type of labor/leisure trade-off. When the income effect outweighs the substitution effect, a worker chooses to consume more leisure and supply less labor. Although impossible in this labor situation, standard economic analysis focuses on the hour-for-hour tradeoff between work and leisure. In the case of professional athletes, leisure may be consumed in the form of reduced effort, especially as it pertains to things such as off-season conditioning, diet, and personal development. To control for the possibility of the income effect, we consider the impact of salary on those who just signed long-term contracts. This is accomplished by an interaction term between SIGNED and the annual salary of the player (SIGNED x SALARY). If effort is adversely affected by the income effect, then we would expect to find that productivity falls with the player's salary. (7)

Each of these independent variables is connected to changes in a player's performance. We consider two measures of a player's productivity in this analysis. The first measure is that which is commonly used by the NBA and media to evaluate players. Denoted [PROD.sub.NBA], this expression of productivity is given in equation (A1) of the appendix. The second measure, developed by Berri (2004), addresses a number of shortcomings of the NBA measure. Specifically, the second measure we employ is designed as a simple estimation of an NBA player's marginal productivity. Designated [PROD.sub.MP] in this study, it is described in detail in equations (A2)-(A5) of the appendix. (8)

Given our choice of dependent and independent variables, the model estimated was:

(2) [DELTA]PROD = [[beta].sub.0] + [[beta].sub.1]D2 + [[beta].sub.2]D12 + [[beta].sub.3] [DELTA]GP + [[beta].sub.4]CEXP + [[beta].sub.5] CWPCT + [[beta].sub.6] [BETA]TMWINS + [[beta].sub.7] [DELTA]ROSTER + [[theta].sub.1] SHIRKING + [epsilon].

Because the relationship we want to investigate is the impact of signing a long-term contract on player productivity, the test for shirking behavior is the one-tailed test that [[theta].sub.1] < 0.

III. EMPIRICAL FINDINGS

The sample used in this study includes those NBA players for whom accurate contract data was available (9) and who received enough playing time for us to establish a legitimate measure of productivity. (10) Altogether, our sample includes 515 observations from the 2000-2001 through the 2002-2003 seasons. Table 1 gives the means and standard deviations for the variable used in this model.

Table 2 contains three sets of ordinary least squares estimates for the coefficients in (2), with the NBA's measure of performance ([PROD.sub.NBA]) employed to measure productivity. The first set considers SIGNED as the shirking variable; the second uses the interaction with contract length (SIGNED x LENGTH); and the final set looks at the interaction with salary (SIGNED x SALARY). One interesting result derived from Table 2 is that the productivity of teammates ([DELTA]TMWINS) is negatively related to a player's productivity. Such a result is consistent with the notion that any one player's performance is subject to a type of diminishing returns in the sense that increases in the overall quality of one's teammates decreases the individual's impact on the team. Furthermore, the dummy variables for experience (D2 and D12) are consistent with our expectations, suggesting that a player's productivity increases over his first two years and declines after he reaches his 12th season. Finally, we see find that injuries are negatively related to productivity in that as games played (GP) falls, productivity falls as well.

In regard to the shirking hypothesis, Table 2 gives weak evidence of a falloff in performance (measured by the NBA method) subsequent to signing a new long-term contract. Although the coefficients on SIGNED and (SIGNED x LENGTH) are significant, the implied impact on productivity is quite small. Evaluated at the point of means, the effect of signing a new long-term contract would reduce productivity by about 2-4%. Finally, in regard to the labor/leisure trade-off, the coefficient on (SIGNED x SALARY) is insignificant, suggesting that increased wealth has no impact on player productivity.

Table 3 contains three sets of estimates for equation (2) when [PROD.sub.MP] is used as the metric for productivity. The important conclusion derived from Table 3 is that long-term contracts do not have any significant impact on this (more sophisticated) measure of productivity. That is, although the estimated coefficients on the shirking variables are negative, they are not significant at any reasonable level. Given the large body of literature concerning the evolution of optimal labor contracts, we do not find this evidence surprising.

One important lesson learned here is that the choice of performance measure has an important effect on the conclusions reached. As noted in the appendix, although the NBA measure is quite popular in the press, it is not highly correlated with team wins. The alternative metric, based on marginal productivity, is exactly the opposite: highly correlated with team wins yet not widely understood. Perhaps our analysis gives us some insight into the debate surrounding the supposed adverse effect of long-term labor contracts on performance. From the economist's perspective, such conduct seems inconsistent with rational and efficient behavior. Yet industry insiders and members of the media insist such activities occur. The evidence presented here suggests that those insisting that players shirk are led to this conclusion by their use of an overly simple method of evaluating players. Our results suggest that shirking does not hold up under a more sophisticated metric of player productivity.

IV. CONCLUDING REMARKS

Prior work on shirking has focused almost exclusively on MLB, typically using slugging average as the measure of player productivity. The current inquiry is the first to examine shirking in the NBA, employing two different measures of player productivity. When the NBA's measure is used, we find evidence consistent with the shirking hypothesis. But when productivity is measured in a fashion more consistent with economists' definition of marginal product, the evidence of shirking evaporates.

At the outset, we noted that the views of economists and industry observers diverged with respect to the subject of shirking. Whereas economists often argued that economic realities of professional sports would make shirking an unlikely outcome, industry insiders and members of the media steadfastly argue that players seem to alter their performance in response to long-term job security. Our analysis suggests that this debate arises primarily out of the metric used to evaluate player productivity. Those insisting on using the NBA's measure of performance can point to the fact that over half (58%) of the players in our sample had a productivity falloff after signing long-term contracts. Yet this simple measure omits a number of important contributions of the player, meaning that some of those accused of shirking may in fact be contributing in other ways. We argue that productivity based on marginal product does a better job of accounting for these missed contributions. In fact, when measured using this metric, less than half (49%) of these players actually underperformed after signing a long-term contract. (11)

ABBREVIATIONS

MLB: Major League Baseball

NBA: National Basketball Association

APPENDIX: MEASURES OF PLAYER PRODUCTIVITY

The aforementioned studies of shirking in professional sports have primarily focused on the sport of baseball. Although this choice may be motivated by the preferences of the individual workers, one should also note that baseball has a number of indices designed to capture the productivity of an individual in a single number. Research employing data from the NBA is generally hampered by the lack of any such index measuring the productivity of a professional basketball player.

Research into the level of racial discrimination in professional basketball reveals that only one measure of productivity, points scored, is consistently correlated with a player's wage. (12)

Other factors of production are not consistently correlated with a player's compensation. Hence one could employ points scored as the sole measure of player productivity. To focus solely on points, though, ignores the many facets of production a player can offer.

The NBA tracks a plethora of measures designed to capture the productivity of an individual player. (13) The NBA uses much of this data in the construction of an efficiency measure, detailed in equation (A1). (14)

(A1) [PROD.sub.NBA] = (PTS + TREB + STL + BLK + AST) - (TO + FGMS + FTMS)

where PTS = points scored, TREB = total rebounds, STL = steals, BLK = blocked shots, AST = assists, TO = turnovers, FGMS = field goals missed, and FTMS = free throws missed.

The NBA's index is quite similar to Dave Heeran's (1992) TENDEX system, (15) Robert Bellotti's Points Created model (see Belloti 1993), (16) and the method IBM employs in ascertaining its player of the year. (17) An apparent problem with these models is that no attempt is made to ascertain the relative value of each statistic. For example, the NBA's model argues that a missed field goal is equal in value to a missed free throw. Consequently, although these indices are easy to construct, the ability of these measures to truly assess worker productivity is not clear.

The seminal work of Gerald Scully (1974) offers an alternative approach. Scully argued that the marginal product of an athlete in a professional team sport could be ascertained by connecting wins to player statistics. (18) Berri (2004) recently adopted the Scully approach in developing a simple measure of marginal product in professional basketball. Berri begins with the following model of team winning percentage (WPCT):

(A2) WPCT = [[alpha].sub.0] + [[alpha].sub.1] *PTS/PE + [[alpha].sub.2] *DPTS/PA + [e.sub.2i]

where PTS/PE = points per possession employed, and DPTS/PA = points surrendered per possession acquired.

In essence, wins in the NBA are determined by how efficiently a team converts its possessions into points, relative to the ability of the team's opponent to elicit points from its possessions. The key to understanding equation (A2) is the concept of team possessions. (19) As defined by Oliver (2003) and Hollinger (2003), the number of possessions a team employs is modeled as:

(A3) PE = FGA - RBO + TO + 0.44*FTA

where FGA = field goals attempted, RBO = offensive rebounds, and FTA = free throw attempts.

As noted by Oliver, a team's possession can end with one of three events: a field goal attempt, (20) a turnover, or some free throws. Equation (A3) recognizes that only a fraction of free throws result in a change of possession. The precise fraction, however, is open to debate. Based on a detailed review of a sample of games, Oliver estimates that ~ 40% of free throws attempted result in a change of possession. Hollinger argues that a more precise measure is 44%, which he claims is the constant that balances best the number of possessions a team employs with those possessions employed by a team's opponent. Hollinger's measure is consistent with the work of Berri, who arrived at the constant of 0.44 by balancing possessions employed and possessions acquired.

The concept of possessions acquired was also introduced by Berri (2004). The purpose of this work was to create a measure of player productivity that assigned a value, in terms of team wins, to each action a player took on the court. Though each element of a team's possessions employed can be traced back to an individual player, several elements of the opponent's possessions employed cannot be connected to any individual player. To overcome this limitation, Berri considered the concept of possessions acquired. As noted by Berri, a team acquires possession (PA) of the ball via the following process:

(A4) PA = DTO + RBD + RBTM + DFGM + 0.44*DFTM,

where DTO = opponent's turnover, RBD = defensive rebound, RBTM = team rebounds, DFGM = opponent's field goal made, and DFTM = opponent's free throw made.

Specifically, equation (A4) implies that a team can acquire the ball after an opponent turnover, the opponent missing a shot that the team rebounds, (21) or the opponent making a shot. As discussed earlier, only a fraction of free throws result in a change of possession.

Table A1 reports the estimation of equation (A2). With this estimation we can now impute the marginal value of much of what each team and player does on the court. By taking the derivative of winning percentage with respect to PTS and PE, one can ascertain the impact of an additional point and possession on team wins, and are reported in Table A2. From this table, we see that each additional point adds 0.033 wins whereas each possession employed subtracts 0.034 victories. A similar calculation produces the marginal impact of DPTS and PA. (22) Consequently the average marginal value of PTS and each of the elements of PE are virtually equal.

Given these values, Table A3 gives the implied value of the statistics tabulated for the players and the team. The team factors include variables that cannot be assigned to individual players. (23) However, the list of player factors (points, field goal attempts, free throw attempts, offensive rebounds, turnovers, steals, (24) and defensive rebounds) can be employed in our evaluation of a player's productivity. (25)

A perusal of the marginal impact of these factors reveals a most pleasant surprise. With the exception of free throw attempts, virtually each of these statistics has an equal impact on wins. Therefore we now have a relatively simple model of player productivity. Given that each factor has virtually the same impact on wins, we can measure player productivity as: (26)

(A5) [PROD.sub.MP] = PTS - FGA - 0.44*FTA + ORB - TO + STL + DRB.

An astute reader might note that (A5) is similar to the measure of productivity employed by the NBA. The crucial difference lies in how shot attempts are valued. The NBA model values equally missed field goals and missed free throws. More important, though, is the impact of made shot attempts. As noted, a made shot results in a change of possession. Therefore, making a shot both moves a team closer to winning the game but also imposes a cost. According to the NBA model, though, a player does not bear the cost of a shot attempt if the attempt is successful. Consequently, according to the NBA's calculations, a player only needs to successfully make 40% of his two-point field goal attempts and 25% of his three-pointers for the benefits of the player's shooting to equal the costs. Most players in the NBA exceed these percentages. Consequently if a player is able to take a large number of his team's shot attempts, the player's value according to the NBA's method can be quite high. In contrast, the method proposed here imposes the cost of each shot attempt, regardless of whether or not the shot is made. As a result, a player must connect on at least 50% of two-point shots and 33% of three pointers. A failure to reach these levels will lead to declines in player value as shot attempts increase.

One should note that the industry and media's evaluation of player performance have been shown to be correlated with rubrics such as the NBA efficiency measure and Bellotti's points created model. Bellotti notes in his work that the media's selection of the most valuable player is virtually always either first or second in the league in points created. Berri and Schmidt (2002) examined the relationship between voting for the All Rookie team, one of the only awards chosen by the coaches, and two measures of player productivity. The first was Points Created, which was shown to explain much of the voting pattern. A second measure, based on the measure of marginal product developed in Berri (1999), was unable to explain the pattern of votes received. Finally, a more recent work by Berri and colleagues (2004) examined player salary via the NBA efficiency measure and the model employed herein to estimate marginal product. The evidence presented suggested that the productivity measure that explained salary the best was the NBA efficiency measure. Explanatory power declined dramatically when the measure of marginal productivity was employed as the primary gauge of player performance.

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(1.) Maxcy (1996), for example, found that long-term contracts tend to be awarded to those players with the most consistent and superior performances across their careers.

(2.) Unlike the National Football League, these contracts are fully guaranteed.

(3.) Notable papers examining the role of baseball managers include Ruggiero et. al. (1997), Horowitz (1994a, 1994b, 1997), Kahn (1993), Scully (1994), and Porter and Seully (1982). Coaching in the National Football League was examined by Hadley et al. (2000). College basketball coaching has been examined by Fizel and D'Itri (1997) and Clement and McCormick (1989). Finally, coaching in the NBA has been previously analyzed by Pfeffer and Davis-Blake (1986).

(4.) This measure follows the work of Idson and Kahane (2000), one of the first studies to consider the impact of team productivity on individual player performance.

(5.) To calculate this factor, one must first note the players who played for the team in both the current and prior season. The minutes of these players were then summed for each season, and this total was then divided by the total number of player minutes in each season. The average of these two numbers was taken to determine the extent of roster turnover on each player's team.

(6.) It is possible that a player attempts to mask his shirking behavior or tries to prove to the public that his lucrative contract was worthwhile by waiting a year before slacking off. If so, then we would expect performance to fall off two years (rather then one) after signing a new long-term contract. We checked for such a two-year lag in shirking behavior and found no evidence using either measure of productivity. We thank an anonymous referee for suggesting this possibility to us.

(7.) We thank Stefan Szymanski for making us aware of this possibility. One must be careful, however, with assigning the increased consumption of leisure to the standard notion of shirking behavior. That is, the traditional notion of shirking as opportunistic behavior arises from the inability to perfectly monitor effort. The issue here is whether the income effect has a measurable effect on the player's consumption of leisure. To the extent that labor contracts internalize this effect, we need to control for greater wealth on the optimal consumption of leisure. Thus, we include the interaction term (SIGNED x SALARY) to control for the income effect on productivity.

(8.) One should note that this measure of productivity is defined on a per minute basis. In other words, like the employment of slugging percentage in the study of shirking in baseball, we are focused on the efficiency of the player.

(9.) The source of these data was USAToday.com.

(10.) Players had to play at least 12 minutes per game and at least 40 games in successive seasons.

(11.) Consistent with Tables 2 and 3, a 58% falloff is weakly significant (but only at the 10% level), and a 49% falloff is not. We thank a referee for pointing out to us this correspondence between the regression results in these tables, and the statistical significance of these two sample proportions.

(12.) Much of the work examining racial discrimination involves regressing a player's wage on measures of productivity and a dummy variable for race. The observation regarding the statistical significance of points scored was made in Berri (2006).

(13.) The productivity data tracked by the NBA that we employ can be found in the 2003-2004 Sporting News: NBA Guide.

(14.) Equation (A1) can be found at the following Web site: www.nba.com/statistics/efficiency.html.

(15.) TENDEX was first formulated by Heeran in 1959. Heeran begins with a model identical to the one currently employed by the NBA but then weights each player's production by both minutes played and the average game pace his team played throughout the season being examined.

(16.) Robert Bellotti's Points Created model, published in 1988, is also quite similar. Bellotti begins with the basic TENDEX model and then simply subtracts 50% of each player's personal fouls. Jenkins (1996) employed the Points Created model in a study of racial discrimination.

(17.) The IBM award differs from the other models by subtracting field goal attempts, rather than missed field goals. The IBM award also ignores the impact of missed free throws, and weights player productivity by both team wins and team statistical productivity.

(18.) Scully's approach was also employed by Medoff (1976), Raimondo (1983), Scott et al. (1985), Zimbalist (1992a, 1992b), Blass (1992), and Berri (1999), among others.

(19.) According to Oliver (2003), the concept of points per possession was noted in Frank McGuire's book Defensive Basketball, published in 1959. Oliver reports that an assistant of McGuire, Dean Smith, utilized the concept of points per possession during his tenure of head coach at the University of North Carolina.

(20.) A field goal attempt will result in a change of possession if it is made or the miss is captured by the opponent. An offensive rebound does not result in a change of possession, hence these must be subtracted from field goal attempts in ascertaining how many times a team has the ball.

(21.) Berri notes that the rebounding of the opponent's missed shots can be separated into two categories, defensive rebounds and team rebounds. The former are rebounds credited to a specific individual. Team rebounds are basically an accounting device. The number of missed shots and rebounds should equal at the end of a game. If a missed shot goes out of bounds, though, a rebound cannot be credited to an individual. Likewise, if a player misses the first of two free throws, no rebound can be recorded for an individual player. Hence, these events are also listed as team rebounds. One should note that only team rebounds that result in a change of possession are included in the estimation of team rebounds that follows. Team rebounds associated with misses on the first of two free throws are not included. For specific details of how team rebounds are calculated one is referred to Berri (2004).

(22.) As one can see from the following the similarity in these values is not a coincidence.

Marginal Value of PTS = [[alpha].sub.1]*[PE.sup.-1]]

Marginal Value of PE = [[alpha].sub.1]*PTS*[PE.sup.-2*] - 1

= [[[alpha].sub.1]*[PE.sup.-1]]*[PTS*[PE.sup.-1]* - 1

The difference between these equations is [PTS/PE]* - 1. The average value of this ratio is 1.02. The calculation of these values used the average number of points and possessions per game. One could also estimate the marginal value of points and possessions for each individual team.

(23.) Following Scott et al. (1985) and Berri (1999), team factors can be used in the evaluation of individual players. Each of these authors allocated variables tracked for the team to individual players according to the number of minutes the player played.

(24.) If the opponent commits a turnover and one can identify the player on the team who is responsible, the identified player is given credit for a steal. So the opponent's turnovers include all of the steals each individual player on the team garners. In Table A3 the value of a steals are listed among statistics tabulated for players while the opponent's turnovers that are not steals (DTO--STL) are listed among team variables. One should also note that a small number of turnovers are not recorded for individual players. One could therefore add team turnovers to the list of team factors in Table A3.

(25.) As noted in Berri (2004), one can employ these marginal values to measure the wins each player creates. In constructing DIFTMWINS, the number of wins each player created was determined. To ascertain the productivity of a player's teammates, the player's estimated win's production and minutes played were subtracted from the corresponding team total. These team values were then employed to measure the per minute productivity of a player's teammates. DIFTMWINS represents the difference between a player's teammates productivity in season t and season t - 1.

(26.) All the player's in the NBA for the 2002-2003 season were evaluated with the simple model listed in equation (A5), which ignores the estimated weights, and via an equation that includes the values listed in Table A3. The correlation coefficient between these two rankings was 0.99. In other words, little is lost when the specific marginal values are ignored.

DAVID J. BERRI and ANTHONY C. KRAUTMANN *

* We thank Stefan Symanski for his helpful comments, and the participants at the 2004 Western Economics Association meetings.

Berri: Associate Professor, Department of Economics, California State University-Bakersfield, 9001 Stockdale Highway, Bakersfield, CA 93311. Phone 1-661-654-2027, Fax 1-661-654-2438, E-mail dberri@csub.edu

Krautmann: Professor, Department of Economics, DePaul University, 1 East Jackson, Chicago, IL 60604. Phone 1-312-362-6176, Fax 1-312-362-5452, E-mail akrautma@depaul.edu

TABLE 1
Summary Statistics

Variable Label Mean SD

Player productivity, [PROD.sub.NBA] 0.448 0.100
NBA model

Player productivity, [PROD.sub.MP] 0.168 0.080
marginal product

Difference in [DELTA][PROD.sub.NBA] -0.004 0.055
[PROD.sub.NBA]

Difference in [DELTA][PROD.sub.MP] -0.001 0.039
[PROD.sub.MP]


Annual salary SALARY 5,460,000 4,407,000

Difference in games [DELTA]GP -1.50 13.8
played

Coaches' winning CWPCT 208.648 265.68462
percent

Coaches' experience CEXP 0.518 5.67912

Difference in team [DELTA]TMWINS 0.025 0.40877
wins

Difference in roster [DELTA]ROSTER 284.733 361.35716
stability

Dummy, two years of D2 0.097 0.29636
experience

Dummy, twelve years or D12 0.117 0.32114
more of experience

Dummy, signed a SIGNED 0.165 0.37159
contract > 3 years

SIGNED crossed with SIGNED x LENGTH 0.810 1.91419
LENGTH

SIGNED crossed with SIGNED x SALARY 1,099,000 2,971,000
SALARY

Number of observations = 515

TABLE 2
Regression Estimates (Dependent Variable: [PROD.sub.NBA];
t-Statistics in Parentheses)

Variable Coefficients Coefficients Coefficients

Constant -0.002 -0.002 -0.003
 (-0.4) (-0.5) (-0.7)

D2 0.031 ** 0.031 ** 0.031 **
 (3.9) (3.9) (3.9)

D12 -0.023 ** -0.024 ** -0.023 **
 (-3.2) (-3.2) (-3.1)

[DELTA]GP 6.2E-04 ** 6.2E-04 ** 6.3E-04 **
 (3.6) (3.6) (3.7)

CEXP 1.6E-04 1.6E-04 1.5E-04
 (0.5) (0.4) (0.4)

CWPCT -3.5E-05 -3.6E-05 -3.6E-05
 (-1.1) (-1.1) (-1.1)

[DELTA]ROSTER 2.1E-05 2.1E-05 2.2E-05
 (0.9) (0.9) (1.0)

[DELTA]TMWINS -0.01 * -0.01 * -0.01 *
 (-1.8) (-1.8) (-1.8)

SIGNED -0.011 ## -- --
 (-1.8)

SIGNED x LENGTH -- -0.002 # --
 (-1.5)

SIGNED x SALARY -- -- -7.8E-10
 (-1.0)

[R.sup.2] 0.10 0.10 0.10

F-statistic 7.41 ** 7.3 ** 7.11 **

* Significant at 10% level.

** Significant at 5% level.

# Significant at 10% level (one-tailed).

## Significant at 5% level (one-tailed).

TABLE 3
Regression Estimates (Dependent Variable: [PROD.sub.MP];
t-Statistics in Parentheses)

Variable Coefficients Coefficients Coefficients

Constant -4.1E-04 -3.5E-04 -6.3E-04
 (-0.1) (-0.1) (-0.2)

D2 0.014 ** 0.014 ** 0.014 **
 (2.4) (2.4) (2.4)

D12 -0.012 ** -0.013 ** -0.012 **
 (-2.3) (-2.4) (-2.1)

[DELTA]GP 3.9E-04 ** 3.9E-04 ** 3.9E-04 **
 (3.1) (3.1) (3.2)

CEXP 1.4E-04 1.5E-04 1.5E-04
 (0.6) (0.6) (0.6)

CWPCT 1.1E-06 5.6E-07 1.4E-07
 (0.1) (0.0) (0.0)

[DELTA]ROSTER 2.8E-07 4.9E-07 1.1E-06
 (0.0) (0.0) (0.1)

[DELTA]TMWINS -0.007 * -0.007 -0.007
 (-1.7) (1.6) (-1.6)

SIGNED -0.005 -- --
 (-1.1)

SIGNED x LENGTH -- -0.001 --
 (-1.1)

SIGNED x SALARY -- -- -5.8E-10
 (-1.0)

[R.sup.2] 0.06 0.05 0.06

F-statistic 3.94 ** 3.97 ** 3.92 **

* Significant at 10% level.

** Significant at 5% level.

TABLE A1
Regression Estimated for Equation (A2) (Dependent
Variable: Winning Percentage; t-Statistics in Parentheses)

Variables Coefficients

Constant 0.495 **
 (5.29)
PTS/PE 3.122 *
 (50.9)
DPTS/PA -3.118 **
 (-48.3)
Adjusted [R.sup.2] 0.947

Observations 286

** Denotes significance at the 1% level.

TABLE A2
Marginal Value of Player and Team Factors

Variable Marginal Value

Points 0.033
Possessions employed (0.034)
Points surrendered (0.033)
Possessions acquired 0.034

TABLE A3
Marginal Value of Player and Team Factors

Factors Marginal Value

Player factors
PTS 0.033
FGA (0.034)
FTA (0.015)
RBO 0.034
TO (0.034)
RBD 0.034
STL 0.034

Team factors
DPTS (0.033)
DFGM 0.034
DFTM 0.015
DTO-STL 0.034
RBTM 0.034