peer-reviewedRaymond A. K. Cox is the Chairman of the Department of Finance and Law, Central Michigan University.  He can be reached at cox1r(@cmich.edu. James Richard Hill is an Associate Professor of Economics at Central Michigan University.

Abstract

We examine whether the "superstar phenomenon," in which a disproportionate share of income accrues to a few earners, can be applied to the National Basketball Association both before and after the 1999 Collective Bargaining Agreement between the owners and players. Results from a log-linear wage regression model and the application of a Yule distribution to wages suggest that the superstar effect applies to both the data from the 1997-1998 season and the 2000-2001 season. However, significant differences in the empirical results suggest that elements of the new agreement are lessening the superstar wage impact.

Introduction

A growing body of evidence supports the “superstar phenomenon,” whereby a few stars in a labor activity outperform all others and reap the lion’s share of the income pool. In professional sports superstar athletes have always been paid more than those with average abilities. The recent introduction of free agency has caused team compensation to become even more concentrated among a few players. After an owner lockout in the National Basketball Association (NBA), a new collective bargaining agreement was reached in 1999 that included sections designed to limit superstar pay and increase compensation for the lower to median income-earner (Hill and Groothuis 2001). Our purpose in this article is to explore the early success of this agreement to alter the superstar phenomenon.

According to Rosen (1981), the superstar phenomenon occurs when small increments in talent are magnified into enormous earnings especially for the top stars. Adler (1985) and MacDonald (1988) continued the development of the superstardom theory. Hamlen (1991) showed the relationship between talent (measured by voice quality) and success for singers (measures by record sales); Chung and Cox (1994) demonstrated the skewed distribution of success (measured by number of gold records). Scully (1974) for baseball and Kahn and Sherer (1988) for basketball, among others, showed the log-linear relationship between a player’s salary and a vector of sports-related skill attributes as well as other variables. We examine the superstar phenomenon in the NBA coupling the standard log-linear regression analysis used in professional sports literature with the approach by Hamlen, and Chung and Cox. We find that a variety of basketball skills possessed by players substantially explain their salaries. Furthermore, the skewed distribution of salaries is compared to a stochastic model of Yule (1924) and Simon (1955) showing a good fit. And because of changes in the collective bargaining agreement between the players and the club owners, the salary distribution has significantly changed to dampen the superstar phenomenon.

A Superstardom Model

Rosen-MacDonald's superstar theory focuses on the accelerating marginal returns to increases in talent. Success is related to talent by the following regression:

(1)

In equation (1)  is the natural log of the player’s salary;  is a constant; Rebounds (Assists, Blocks, Points) is the rebounds (Assists, Blocks, Points) per game: Draft is the draft number of each player’s selection in the NBA draft; Years is the number of years played in the NBA; and  is the square of the number of years played in the NBA. Rebounds, Assists, Blocks and Points are calculated as the average statistic for the previous four seasons. The sign of the relationship is expected to be the positive for all the variables except for  and Draft. The coefficient of  should be negative as this variable captures the concavity of salary  versus years: productivity declines with age in physical sports. The coefficient of Draft is anticipated to be negative as players with great promise displayed by previous success in college are drafted earlier with a lower number (pick number 1, then 2 and so on).

The stochastic model derived by Yule (1924) and further elucidated by Simon (1955) is used to fit the NBA salary distribution. A diverse collection of sociological, biological, and economic phenomena is propelled by certain probability mechanisms that produce a class of (Yule) distributions like negative binomial or log series distributions. Diverse phenomena that can be modeled by this same stochastic process include, e.g. distributions of word sample by their frequency of occurrence (Thorndike 1937; Good, 1953), of scientists by number of papers published (Leavens 1953), of cities by population (Zipf 1949), of incomes ranked by size (Champernowne, 1953), of biological genera by number of species (Yule 1924), and of singers by number of gold records (Chung and Cox 1994).

The Yule distribution is:

(2)

,                                                                              

where and  are constants and is the beta function of and 

(3)

,

A derivation of the above Yule distribution can be found in Yule (1924), Simon (1955), and Chung and Cox (1994).

Given the two assumptions of the stochastic model generating the Yule distributions:  ,                                                                                       

(4)

Where ,in the context of this study, is the proportion of NBA player with an income in the $1 million range,

is the total number of players, and is the constant probability of a player earning the average salary.

Furthermore, the beta function is the transformed:

(5)

and,

                                                                                           

(6)

Where is the proportion of a player in the million dollar range ( is in increments of approximately one million dollar amounts).

The Yule Model prediction of the percentage of players earning salaries in one million dollar increments is shown in the far right hand column of Table 2.

Empirical Results

Data Description

Our salary data are compiled and reported by Bender online and originally came from newspaper articles, typically USA Today. These two seasons were chosen as a before and after snapshot of the NBA undergoing negotiated salary restrictions between the players and management. The statistics of player performance are from the third edition of Official NBA Encyclopedia and the 1997-1998 and 2000-2001 editions of Sporting News Official NBA Register.

Empirical Testing.

Table 1 shows the regression results of the talent measures explaining both the level and the natural log of the player’s NBA salary in each of the two years. The results indicate the superiority of the log-linear format in terms of explanatory power, particularly for the earlier time period. All the variables are correctly signed and all coefficients in the log-linear format are statistically significant as measured by their t-statistic. Clearly, performance translates into higher salary, i.e., talent begets income.

Table 1

Empirical Estimates

                                              Independent Variable             Dependent Variable

                                                          Coefficients (t-statistic in brackets)

                                                          1997-98                                     2000-01

                                                LNSAL             SAL              LNSAL                SAL

            Constant                    13.133            -1057208           13.061           -2139962

                                                (112.598)       (-2.730)              (120.194)       (-5.070)

            Rebounds                  0.07891          115724.84         0.118              418615.12

                                                (3.709)            (1.638)                5.896)             (5.382)

            Assists                       0.07509          57531.35           0.143              590847.28

                                                (3.047)            (0.703)                (7.704)            (8.172)

            Blocks                         0.264              1066771.1         0.320              1886227.9

                                                (3.051)            (3.71)                  (3.945)            (5.983)

            Points                         0.04734          204646.47         0.02484          111029.47

                                                (4.559)            (5.936)                (4.443)            (5.113)

            Draft*                          -0.01153        -7448.03            -.007206        -12646.66

                                                (-6.417)          (-1.249)              (-4.687)          (-2.117)

            Years                          0.126              214892.06         0.210              402680.32

                                                (4.207)            (2.169)               (8.071)            (3.992)

            Yrssq                          -0.007819      -10132.84          -0.01154        -20991.13

                                                (-4.133)          (-1.613)              (-6.899)          (-3.230)

             n size                          371                 371                     385                 385

            F-statistic                   85.470            49.171                103.486          91.333

Adjusted R2                           61.4%             47.6%                 65.1%             62.2%

Notes:  Rookies are deleted from the sample since they have no prior measured NBA skill set. *The value for draft number was set equal to 60 for players who were not drafted but made it into the NBA from the 1989 season onward. This value was chosen because the draft was limited to two rounds after this date. Beginning in 1995 there were 29 franchises, so with two rounds this meant the last player drafted had a draft number of 58.

Table 2 presents the evidence of the Yule distribution fitting the distribution of the player salaries in the NBA. To test whether the Yule distribution describes the NBA salary distribution we conduct the Chi-square goodness-of-fit test comparing the actual number of players in each category to the predicted number using the Yule percentages.

 

Table 2

Frequency Distribution of NBA Players by Salary for 1997-98 and 2000-01

($million)

                                                                                                                           Yule

                                                                                                                        Prediction

                                                                                                                         Salary       

                 Actual Number of Players  Percentage of Players         Percentage

                                                                       (Millions)

                         1997-98         2000-01         1997-98         2000-01              p = 1         

  1                      241                 196               59.65              44.34                50.00

  2                        59                    90                14.60              20.36                16.67

  3                        42                    39                10.40                8.824                8.33

  4                        23                    23                  5.69                5.20                  5.00

  5                        17                    25                  4.21                5.66                  3.33

  6                          3                      7                  0.74                1.58                  2.38

  7                          3                      8                  0.74                1.81                  1.79

  8                          4                    11                  0.99                2.49                  1.39

  9                          3                      9                  0.74                2.04                  1.11

10                          1                    15                  0.25                3.39                  0.91

11                          3                      5                  0.74                1.13                  0.76

12                          1                      4                  0.25                0.90                  0.64

13                          1                      2                  0.25                0.45                  0.55

14                          1                      3                  0.25                0.68                 0.48

15                          0                      1                  0                      0.23                  0.42

16                          0                      2                  0                      0.45                  0.37

17                          0                      0                  0                      0                        0.33

18                          0                      0                  0                      0                        0.29

19                          0                      2                  0                      0.45                  0.26

20                          1                      0                  0.25                0                        0.24

33                          1                      0                  0.25                0                        0.08

Total                 404                 442

As the Chi-square statistics are less than the 299.5 critical values of 20.28 and 21.96 respectively we cannot reject the hypothesis that the Yule distribution with  = 1 depicts the stochastic process underlying the superstar phenomenon in the NBA.

To examine whether the NBA salary distribution has changed between 1997-98 and 2000-01, a chi-square statistic was calculated using the former and latter period as the predicted and actual, respectively. The Q is 28.66 with a 299.5 of 14.86 with degrees of freedom of 4. Clearly, there has been a radical change in the NBA salary distribution between these time periods. However, given the results from the Yule distribution the salary distribution continued to be highly skewed and consistent with that of the superstardom theory.

A comparison of the regression results from the two salary sample years indicates various changes in coefficients. However, results of Chow Tests for the log-linear format, indicate that only the coefficients of Points, Assists, and Years have undergone significant change. The coefficient of Years increased; this suggests that changes in the new NBA agreement that require minimum pay increases with years of service may have increased the seniority pay component of the compensation system that is generally a cornerstone of unionized firms. Previous evidence from baseball (Hill and Spellman 1983) indicated that free agency lowered seniority pay for players. The coefficient of Points decreased while the coefficient of Assists increased. These changes could indicate an apparent shift in compensation toward "team players" and away from superstar scorers.

Summary and Conclusions

A cursory inspection of NBA player salary distributions indicates a noticeable skewness. The pattern of extremely high salaries for only a few has been labeled the superstar phenomenon.

We attempted to provide empirical evidence of the superstar theory as promulgated by Rosen and MacDonald. First, empirical results show that greater talent, in playing basketball in the NBA, commands greater salary (success). Second, the NBA salary distribution is fitted to the stochastic process that describes a variety of sociological, biological, and economic phenomena as described by the Yule distribution. The empirical results indicate the Yule distribution is the same model as the NBA salary distribution for both time periods, before and after the changes in the NBA collective bargaining agreement. However, results also indicate a significant change in the compensation models between periods and a lessening of the superstar phenomenon.

Appendix A

Top 20 Salaries

Name                         Amount                                 Name                         Amount

Michael Jordan          $33,140,000                          Kevin Garnett $19,610,000

Patrick Ewing            $20,500,000                          Shaquille O’Neal      $19,285,715

Horace Grant             $14,285,714                          Alonzo Mourning      $16,880,000

Dennis Rodman        $13,500,000                          Juwan Howard          $16,875,000

Shaquille O’Neal       $12,857,143                          Hakeem Olajuwon    $16,700,000

David Robinson        $12,397,440                          Karl Malone               $15,750,000

Alonzo Mourning       $11,254,800                          David Robinson        $14,700,000

Juwan Howard           $11,250,000                         Dikembe Mutombo   $14,400,000

Hakeem Olajuwon     $11,156,000                         Patrick Ewing            $14,000,000

Gary Payton               $10,514,688                          Jayson Williams        $13,800,000

Dikembe Mutombo   $  9,615,187                          Scottie Pippen          $13,750,000

Reggie Miller             $  9,031,850                          Rasheed Wallace     $12,600,000

Chris Webber            $  9,000,000                          Gary Payton               $12,200,000

Shawn Kemp             $  8,600,000                          Tim Hardaway           $12,000,000

Larry Johnson            $  8,460,714                          Chris Webber            $12,000,000

Derrick Coleman       $  8,002,800                          Shawn Kemp             $11,720,000

Kevin Johnson           $  8,000,000                          Arvydas Sabonis       $11,250,000

Latrell Sprewell          $  7,770,000                          Damon Stoudamire  $11,250,000

Anfernee Hardaway  $  7,580,000                          Larry Johnson            $11,000,000

Elden Campbell        $ 7,000,000                           John Stockton            $11,000,000

Works Cited

Adler, M. “Stardom and Talent.” American Economic Review 75 (March 1985): 208-212.

Bender, P. Patricia's Various Basketball Stuff.” 6 Jan. 2004 <http://www.dfw.net/~patricia>.

Champernowne, D. G. “A Model of Income Distribution.” Economic Journal 63 (June 1953): 318-351.

Chung, K. H., and R. A. K. Cox. “A Stochastic Model of Superstardom’s An Application of the Yule Distribution.” Review of Economics and Statistics 76 (November 1994): 771-775.

Good, I. J. “The Population Frequencies of Species and the Estimation of Population Parameters.” Biometrika 40 (December 1953): 237-264.

Hamlen, W. A. Jr. “Superstardom in Popular Music: Empirical Evidence.” Review of Economics and Statistics 73 (November 1991): 729-733.

Hill, J. R., and P. A. Groothuis. “The New NBA Collective Bargaining Agreement, The Median Voter Model and a Robin Hood Rent Redistribution.” Journal of Sports Economics 2.2 (2001): 131-144.

Hill, J. R., and W. Spellman. “Professional Baseball: The Reserve Clause and Salary Structure.” Industrial Relations 22 (Winter 1983): 1-19.

Kahn, L. M., and P. D. Sherer. “Racial Differences in Professional Basketball Players’ Compensation.” Journal of Labor Economics 6 (January 1988): 40-61.

Leavens, D. H. “Communications.” Econometrica 21 (October 1953): 630-632.

MacDonald, G. M. “The Economics of Rising Stars.” American Economic Review 78 (March 1988): 155-167.

Rosen, S. “The Economics of Super-Stars.” American Economic Review 71 (December 1981): 845-858.

Scully, G. “Pay and Performance in Major League Baseball.” American Economic Review 64 (December 1974): 915-930.

Simon, H. A. “On a Class of Skew Distribution Functions.” Biometrika 42 (December 1955): 425-440.

Thorndike, E. L. “On the Number of Words in Any Given Frequency of Use.” Psychological Record 1 (1937): 399- 406.

Yule, G. U. “A Mathematical Theory of Evolution, Based on the Conclusions of Dr. J. C. Willis, F.R.S.” Philosophical Transactions of the Royal Society B 213 (1924): 21-87.

Zipf, G. K. Human Behavior and the Principle of Least Effort. New York: Addison-Wesley Press, 1949.

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