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PAGE 1 PAY AND PERFORMANCE IN PROFESSIONAL FOOTBALL: AN ANALYSIS OF MARGINAL REVE NUE PRODUCT AND MONOPSONISTIC EXPLOITATION IN THE NATIONAL FOOTBALL LEAGUE BY NEAL HARGIS A Thesis Submitted to the Division of Social Sciences New College of Florida in partial fulfillment of the requirements for the degree Bachelor of Arts Under the sponsorship of Professor Richard D. Coe Sarasota, Florida May, 2011 PAGE 2 A cknowledgements I would like to thank my pare nts for all of your support over the years. I could not be where I am without all of your positive advi ce and help shaping me into the individual that I currently am. I know the task has not been easy for you. I would like thank all of my committee me mbers Professors Richard Coe, Tarron Khemraj, and Duff Cooper for aiding me in this incredibly challenging and arduous process. Next I would like to thank all of my friends at New College for their motivation when I needed it most. You guys supported me even in times when I thought finishing would be impossible. I would like to thank Jasmine Zeki and Alyssa Sonchaiwanich in particular for helping so much with my editing. Most importantly, I would like to thank my amazing sister Jacqueline Hargis for everything she has helped me through in my lif e. I would have never been able to make it through this process without my sister's guidance and discipline. My sister is the most remarkable person that I know and my role model. ii PAGE 3 Table of Contents Acknowledgements ii Figures v Abstract vi Introduction Score and purpose of the investigation 1 Method of Study 4 Chapter 1: Theoretical and Literature Background Monopsonistic Labor Market Structure 6 Perfectly Competitive Labor Market Structure 8 Literature Review i) Pay and Performance in Major League Baseball 11 ii) Pay and Performance in Major League Baseball: The case of the first family of free agents 15 iii) Baseball and Billions 16 iv) What's Wrong with ScullyEstimates of A player's Marginal Revenue Product 18 v) Salary Vs. Marginal Revenue Product Under Monopsony and Competition: The case of Professional Basketball 21 vi) Pay, Performance, and Competitive Balance in the National Hockey League 24 vii) Measuring Managerial Efficiency: Th e Case of Baseball 26 Chapter 2: Methods Regression Analysis 28 Quarterback Performance Regression 29 Running Back Performance Regression 32 Wide Receiver Performance Regression 34 Defensive Linemen Performance Regression 35 Linebacker Performance Regressi on 38 Secondary Performance Regression 39 Regressions Relating Winning Percentage to Revenue 40 Relating Gate Revenue to Winning 41 Calculating Marginal Revenue Product 42 Calculating Monopsonistic Exploita tion 43 Chapter 3: Results Position Performance Quarterbacks 45 Running Backs 46 Wide Receiver s 47 iii PAGE 4 Defensive Linemen 48 Linebackers 50 Secondary 51 Relating Revenue to Winning 52 Relating Gate Revenue to Winning 54 Monopsonistic Exploitation Quarterbacks 55 Running Backs 56 Wide Receiver s 58 Defensive Linemen 59 Linebackers 60 Secondary 61 Results for Marginal Gate Revenue 63 Chapter 4: Conclusion Evaluating Results 65 Improvements to the Analysis 68 Quarterbacks 69 Running Backs 70 Wide Receivers 72 Defensive Linemen and Linebackers 73 Secondary 74 Discussion 76 References 77 iv PAGE 5 Figures and Tables Figure 11 7 Figure 12 10 Figure 31 53 Figure 32 56 Figure 33 57 Figure 34 58 Figure 35 59 Figure 36 60 Figure 37 62 Table 31 63 Table 32 64 v PAGE 6 vi PAY AND PERFORMANCE IN PROFESSIONAL FOOTBALL: AN ANALYSIS OF MARGINAL REVE NUE PRODUCT AND MONOPSONISTIC EXPLOITATION IN THE NATIONAL FOOTBALL LEAGUE Neal Hargis New College of Florida, 2011 ABSTRACT This thesis examines the microeconomic structure of th e National Football League as it pertains to player compensation. Building off previous work in this field, this work establishes a value for professional football player marginal revenue product by relating on field performances of six differe nt positions to game outcomes, winning percentages, and team revenues using econometri c and statistical tools. In addition, this work is able to quantify a va riable that evaluates performa nces for football's running back and wide receiver positions. These determinat ions were compared to player's actual salaries and used to measure a rate of monopsonistic exploitation in the National Football League. The calculated rate of monopsonistic exploitation answers a lingering question as to whether or not professional football play ers are paid fair salaries equal to their marginal revenue products. Professor Richard D. Coe Department of Social Sciences PAGE 7 Introductio n Scope and Purpose of the Analysis The professional sports business has e volved over the past century and a half. Player compensation, an integral issue in sports, leads to countless labor struggles including lock outs and player strikes. Ev ery major professional sport in America has gone through similar struggles in order to shape themselves in to their current respective league structures. The methods of dete rmining player compensation today were developed to incorporate collective bargai ning between owners and players. These collective agreements shape the overall struct ure of player compen sation including rookie pay, minimum salaries, salary caps, and percen tage of revenue to be devoted to salary (NFL Collective Bargaining Agreement, 2006). Player salaries have increased over time as a percentage of league total revenue. Currently, the amount of compensation that athletes receive is likely to be much closer to their true margin al revenue product. Marginal revenue product is loosely defined as the cha nge in revenue that a firm takes in as the result of employing an addi tional unit of labor (C ase and Fair, 2004). Marginal revenue product related to professional player compensation will be examined as the amount by which an athlete's contributions affect his team's ability to win, leading to increased revenue. Marginal revenue product is the appropriate measure to determine how much an individual athlete helps his team earn revenue. Generally, in sports, revenue is generated by winning. Winning results in incr eased game attendance, in addition to an increased television audience. Teams earn lucr ative profits through their gate revenue and selling television broadcasti ng rights (Scully 1974). The National Football League (NFL) stands at the pinnacle of all major American professional sports leagues. The owners a nd administration of the NFL have formed a 1 PAGE 8 collaborative league structure related to negotiating, com petitive balance, and revenue equality. These different controls ensure competitive and financial balance in the NFL (Fort, 2006). The NFL negotiate s with outside sources, such as TV broadcasters and apparel companies, as an entire unit rath er than on a team by team basis. Revenue brought in from these many arrangements is shared amongst the 32 NFL franchises, which promotes their financial balance goals Notably, the NFL has a higher percentage of gate sharing than any other major pr ofessional sports league in the country1 (Fort 2006). The NFL employs some of the finest athletes in the world, but many people believe that these players are overpaid. One ra tionality is that teachers have a greater influence on society and deserve to be paid more. The explanation has to do with their personal scale of operations. Personal scale of operations refers to an individual's gross output and the dollar value of his or her abil ity (Mayer, 1960). A teacher can only have an effect on the students in his or her classroom, while a professiona l football player has the ability to entertain millions. Therefore, a teacher's contributions could be more valuable to each individual, but they cannot add up to the amount that a player contributes through his volume of business (Rosen and Sanderson, 2000). One motivation for this research is that many still believe athletes are overpaid. While it is quite possible that they are overpaid, history supports the alte rnative. There have been numerous studies using a combination of statis tics and econometrics showing th at athletes are actually being paid below their marginal revenue product. 1 For future reference, the other major professional sports leagues will refer to the National Basketball Association, Major League Baseball, and the National Hockey League. 2 PAGE 9 This work quantifies the marginal revenue product for professional football players in order to calculate a rate of monopsonistic exploitation occurring in the NFL. In 1974, sports economist Gerald Scully (1974) was the first to publish this type of an analysis. Scully attempted to quantify th e marginal revenue product of professional baseball players by using regressions employing their onfield statistics and team revenue. He was motivated by the reserve clause in ba seball, which effectively restricted player mobility to different teams and prevented athletes from negotiating their contracts without being unilaterally dominated by the ow ners with whom they were negotiating. The motivation for this thesis, which will fo cus on professional footba ll, pertains to the current collective bargaining agreement negotia tions between the owners and players in the NFL. In order to calcu late monopsonistic exploitation in the NFL, the process was broken down in steps using economet rics and statistics. The firs t step in the analysis was to figure out a proper evaluation for an athlet e's performance. The players measured were divided into six different po sitions: quarterbacks, runni ng backs, wide receivers, defensive linemen, linebackers, and secondary players These players were evaluated using variables developed by aggregating their standard onfield statistics. For quarterbacks (QB), their published QB rati ng was used (NFL Enterprises LLC, 2009). Running backs (RB) and wide receivers (WR) we re evaluated with a ra ting created in this work using a process similar to data envel opment analysis. The three defensive positions were evaluated using an innova tive statistic that relates the amount that a player contributes to a play to the probability of th eir team winning that game. This statistic is titled winning probability added (WPA). 3 PAGE 10 Once the players' performances were evaluated, the next step in the analysis was to run regressions tracking the interaction of player performance to the outcome of the game. Regressions were also run to relate the amount that winning effects team revenue. The coefficients produced in these regressions were used to calculate the marginal revenue product of the six positions. The last step of the analysis was to relate these calculated marginal revenue products to actual salary in order to determine the rate of monopsonistic exploitation in the NFL. The present research seeks to answ er the question of whether or not monopsonistic exploitation is occurring in th e professional football labor market. This topic is relevant towards the current co llective bargaining negotiations between the player's union and the NFL. These negotiatio ns have been through several different stages, including federal arbitration, decertif ication of the player's union, and lawsuits currently in progress. The current court cas e's ruling will provide the winner with a significant bargaining chip once the two sides return to negotiations. The present results could provide an asset to negotiations proving that players ar e being exploited. Additionally, it could encourage the owners to adjust the level of revenue allocated toward player costs. Method of Study Chapter I introduces microeconomic m odels detailing the theory behind both monopsonistic and perfect competition labor market structures. Following the theory explanation, a review of prior methods used to calculate the margin al revenue product of athletes from various sports is included. Ma rginal revenue product will be discussed for baseball, basketball, and hockey players as well as baseball managers. The article 4 PAGE 11 published b y Scully (1974) will be the basis of the discussion as his methods pioneered the process behind the various analyses. Chapter II of this thesis builds on the methods discusse d in the previous chapter and introduces the process used in the analys is. The econometric and statistical tools to be used in the calculations will also be discussed in regards to the theory behind them and their necessity to the pr ocess. As a part of this discu ssion, the calculatio ns' expectations will be forecasted and explained. Chapter III looks at the results of the regressions and calculations using econometric theory to explain the values ge nerated. Chapter IV will discuss the results and how they could affect negotiations betw een the players and ow ners, in addition to some aspects of the analysis that could im prove the process of de termining a value for NFL players. Chapter IV will conclude by alluding to topics of further research in the area of studying the economic structure of the NFL. 5 PAGE 12 Chapter 1: Theoretical and Literature Background Monopsonistic Labor Market Structure A m onopsonistic market is similar to the microeconomic structure of a monopoly. They both represent situations of imperfect competition as individual firms are able to control prices in their respective markets. A monopsonistic market structure is ma de up of one buyer with multiple sellers and behaves very similar to a monopoly. The standard area where a monopsonist operates occurs in a firm's factor market where they wi ll be the sole of buyer of a given factor of production. If the market was competitive than the firm would be forced to purchase the necessary amount of the factor at the given market price, however when they are the sole buyer of the factor, the monopsoni st is able to dictate the price, becoming a price maker (Varian, 2006). Figure 11 shows the method in which a monopsonist is able to minimize their input costs in the labor market to maximize profits. 6 PAGE 13 Figure 11 Monopsonistic Micro economic Model (Graphical representation of a monopsonistic labor market) The profit maximizing condition for a monopsonist in the labor market is to set the marginal revenue product of labor equal to the marginal co st of labor. This represents the monopsonistic equilibrium amount of labor that the firm will employ. The equilibrium levels of wage and labor in this model are represented by the points WM and LM, respectively. The equilibrium levels of wage and labor for perfect competition are represented by the points WP and LP, respectively. The monopsonist is able to save additional costs due to exploita tion of wages represented by the green area in Figure 11. Areas A and B in Figure 11 represent th e two types of deadweight loss occurring as a result of the monopsonist. Area A represents the consumer surplus lost as restricting 7 PAGE 14 the am ount of labor employed prevents the monopsonist from employing additional labor. Area B represents the loss of producer surplus due to labor willing to work for the perfectly competitive equilibrium wage, but not willing to work for the monopsonist's decreased wage (Varian, 2006). Viewing the NFL as the monopsonist in th is circumstance, area B would be much smaller as there are less athletes refusing to play in the NFL due to wages being too small. Area A would represent the majority of the d eadweight loss to society since many more athletes could be employed if the NFL increased the number of teams in the league, allowing for more j obs as players. WP represents the marginal revenue product of labor that this thesis will examine and calculate. WM represents the curren t level of wages in the NFL provided that the league acts as a monopsonist in the labor market. Perfectly Competitive Labor Market Structure An integral part of NeoClassical Economic thought is the assumption of a perfectly competitive market. The theory behind a perfectly competitive labor market precludes several assumptions by extension. At the aggregate level these assumptions include numerous firms vying for labor avai lable and many qualified sources of labor available. Additionally, perfect information for both parties is assumed, as well as neither party having the ability to control market wa ges or restrict labor mobility (Krugman and Wells, 2005). To view these assumptions in the microe conomic aspect of a professional football league proposes issues. Firstly, there is no assumed collusi on among the various teams in the league as they are competing with each other to employ various players. An issue arises with this assumption as teams do not act entirely as separate entities in their 8 PAGE 15 m ethods of acquiring labor. One of the most integral issues where professional football doesn't support the assumptions of a perfectly competitive labor market is the idea of homogenous labor. Since every football player varies greatly from one another, the idea of a homogenous labor force can not be accepted. Even considering labor market segmentation where the labor force for a give n industry is divided into different subdivisions, the idea of a homogenous labor fo rce can not be accepted. These subdivisions in the industry of professional football would be represented by the various positions in the game. In a perfectly competitive labor market, firms are price takers in regards to wages. Figure 12 represents various st ructures of perfectly competitive labor markets pertaining to different possibilities in the market structure of professional football. 9 PAGE 16 Figure 12 (Graphical representation of three different perfectly competitive labor markets) Figure 12(a) represents a standard economy with regular downward sloping demand and upward sloping labor supply curve. In labor markets, demand is defined as the marginal revenue product of labor. The market in Figure 12(a) shows similar elasticity for both supply and demand of labor. Figure 12(b) represents a labor perfectly competitive labor market with a perfectly elastic supply curve. The firm is able to hire as much labor as they see fit at the given market price of W bar. If this model we re to fit the NFL, it would mean that there is an unlimited supply of football players who all work for a given wage. If this occurred in the NFL, there would likely be labor segmentation to design a set wage by each position. This wage could be the average of all salaries for each position. Figure 12(c) represents a labor market with a perfectly inelastic supply of labor fixed at the Q bar level. In this situation, the amount of labor demanded determines the wage of labor. Increasing (decr easing) demand would increase (decrease) the wage rate. An ideological market with this supply curv e in the football industry would mean that there are a limited number of players with the talent to pl ay professional football, and these players will be paid their marginal re venue product W*. The only variable in this 10 PAGE 17 m arket would be the quantity of labor th at the firm would em ploy. As long as their marginal revenue products are high, then they should be paid accordingly. The primary difference between the monopsonist model and these perfectly competitive models are the equilibrium wage rates. In the monopsonist model, the players will be paid below their marginal revenue products and the difference between this and their actual pay would represent the amount of monopsonistic exploitation. In a perfectly competitive market, players will be paid their margin al revenue products (Downward, Dawson, and Dejonghe, 2009). Literature Review i) Pay and Performance in Major League Baseball The literature that genera ted the ideas and methodologies behind this thesis will be reviewed in the following section. Gerald Scully has been regarded as the first person to establish a marginal revenue product fi gure for professional at hletes. His methods played a role in the collective bargaining ne gotiations that helped to end the reserve clause in baseball. The rese rve clause was initially put into baseball in 1889, and was enacted by the owners in the National League in order to help protect their interests in terms of acquiring and retaining players (For t, 2004). Contracts at this time only lasted one year with salary being their only major va riable. The reserve clause stated that at the end of each season the owner's of the team had th e right to retain the talents of the player for the next year (Fort, 2004). The salary that an owner offered the players was nonnegotiable. The only options that players had with the reserve clause were to refuse the offer and request a trade, or sit out the entire season. 11 PAGE 18 Scully sought to establish a figure for marginal revenue product to help prove that the reserve clause caused a high degr ee of monopsonistic ex ploitation favoring the owners. The problem which inspired Scully's ar ticle was that the reserve clause prevented free mobility of labor causing the players to be exploited by the owners. Baseball's owners were in collusion to keep the labor situation from changing. The players' goal was to break apart the owners by making them co mpetitive towards each other in seeking out the best talent to generate the most wins (Scully, 1974). In order to break the owners' bond, the players needed significant empirical re sults showing the degree of which they were being taken advantage, and economica lly rational figures showing what their salaries should amount too. Scully introduces his model by calling it, "A simple Model of Marginal Revenue Product and Salary Determination in Major League Baseball." The term "simple" to describe these models seems necessary to in clude in any sort of evaluation for player marginal revenue product, especi ally based on performance. In any process such as this, it is impossible to quantify and account for ev erything that goes into a team winning and a player's contributions towards success. Scully's first step was to establish an equation for team victories based on the assumption that winning is the method by which gross revenues are generated. He used winning per centage as the endogenous variable, and his exogenous variables included a v ector consisting of player sk ills and a vector consisting of other input factors such as coaches and ot her intangibles. Scully's revenue equation is much more involved as he wanted to account for many factors that could have affected attendance such as: market size, racial di scrimination, and whether a team was in the National League or the American League. Scully notes that at this time National League 12 PAGE 19 play was regarded as superior to play in th e American Leag ue, and also that teams with higher percentages of AfricanAmerican s had lower revenues (Scully, 1974). The first half of the revenue equation was gate revenues, while the second half was broadcasting revenues. Scully's cost f unction consisted of a summation of player salaries based on representations of the player s' various skill levels and a summation of the nonplayer input costs. For his estimations of player marginal revenue product, Scully divided up baseball players into two types: hitters and pitche rs. His performance variable for hitters was slugging average which is deri ved by dividing a player's total number of bases earned through hitting, by their total numbe r of at bats (Somme rs & Quinton, 1982). While being an accurate measure for hitting performance, slugging average does not take into account the defensive aspect of a player's performance. For pitchers, Scully used the team "strikeouttowalk" statistic, which he claimed was the best method for measuring pure pitching performance. He also include s two exogenous dummy variables that detail games which have been played when a team is contending for the playoffs versus when they have already been eliminated. These va riables are applicable as team performance diminishes once they have been eliminated from playoff contention. His results at this stage were all statistically significant to the =1 percent level with a high R2 value, meaning that the endogenous variable was mostly correlated w ith the parameters measured. Scully's data was recorded on the te am level rather than the individual, so he divided the coefficients produced in his resu lts by the number of hitt ers and pitchers on the team in order to quantify an individual's performance. In most of these types of analyses, a ccounting for revenues and their many input factors is more difficult than the evaluation of player performance. In order to capture as 13 PAGE 20 m any of these factors to revenue as possible, Scully included: winning percentage, market size, team attendance, NL dummy, a stadium dummy differentiating between old and new stadiums, and the racial discrimination factor. Once again, all of his coefficients were significant at better than the one percen t level. The most important part of these results was that his winning percentage variable was significan t, and he could use it to determine marginal revenue product for the individual athlete by multiplying the coefficient in front of winning percent with the coefficient in front of the performance factors. This allowed him to put a monetary value on every percentage point increase in statistics such as slugging average and strikeouts to walks. When applying the team data to the individual data, one assumption Scul ly was forced to make was that, "team performance is simply the linear summation of individual performance" (Scully, 1974), which means that he was not able to include interplayer relations, such as teamwork and chemistry, in his results. While Scully coul d have calculated each individual athlete, he decided to divide them up in to average, above average, and below average players. Through this, he developed gross MRP figures He then subtracted his estimation of player costs through training and coaching from his gross MRP figures to develop his net MRP estimations. Scully's table comparing his MRP results w ith actual salaries showed that players were being paid less than their MRPs to a ve ry high degree. This implied that there was extensive monopsonistic exploita tion occurring, which he went on to show in his second table. Scully's policy suggestions started w ith the reserve clause being eliminated and replaced with a different system that restricts player mobility less. One of his suggestions was a policy similar to the NFL's "Rozelle Rule", which included compensation in the 14 PAGE 21 for m of draft picks to a team giving away a fr ee agent. Scully's article was a criticism of the reserve clause and empirically showed that it allowed for large degrees of monopsonistic exploitation in the MLB. In 1975, a year after this article was published, the MLB refigured its reserve clause as a result of collective bargaining agreements. Therefore, it would appear that Scully le ft a mark on baseball through his empirical results establishing MRP for prof essional baseball players. ii) Pay and Performance in Major League Baseball: The case of the first family of free agents Paul M. Sommers and Noel Quinton's (1982) publication was written as a reaction to the process that Scully pioneered. Using Scully's models, Sommers and Quinton looked at the contracts that re sulted in the adjustment of th e reserve clause and whether or not the initial free agents were being paid their marginal revenue product. The modeling techniques used in this article were basically the same as Scully's methods. One difference was that this model employed a pa rameter of whether or not a team was an expansion club, which was added to the winni ng percentage equation. Generally when a team plays its first year in a major sports league, they do not have a very successful record. Also, instead of usi ng Scully's factors for teams slugging average (TSA) and team strikeouts to walks (TSW), this model altered them a bit to be the ratio of TSA and TSW when compared to the league average. This makes the parameters s how, "relative rather than absolute strength" of performance to their peers (S ommers and Quinton, 1982). The parameters in the revenue equation were also very similar, except this equation included expansion teams and a dummy that noted if a team shared a stadium with another. These regressions were calculated to see whether or not the fourteen free agents that arose from 15 PAGE 22 the reform ation of the reserve clause actu ally managed to earn their salary when compared to their MRP. Earning their salaries pertains to thei r ability to meet their team's expectations based on their contributions towards winning and increasing attendance. In Sommers and Quinton's table they show that thirteen of the fourteen free agents succeeded in earning their salaries by havi ng a higher Gross Marginal Revenue Product than their annual salaries In addition, their annual contract cost was calculated, which incorporated other costs that arose from the contract such as bonuses and insurance. In this case, eight of the fourteen free agen ts managed to have higher marginal revenue products than their newly calcu lated input costs. Another interesting result was that the free agents, while generally getting paid be low their marginal revenue products, didn't exceed the rest of players on the team enough in performance to make the net cost for acquiring them exceed their net benefit. They go on to say that, "these results suggest that free agents were not so much overpaid as other ballplayers were underpaid." (Sommers and Quinton, 1982). This article showed that removal of the reserve clause helped salaries rise closer to players' MRPs. The authors conclude by disc ussing that the next generation of free agents was much larger, and were also paid close to their marginal revenue products, showi ng that the removal of the reserv e clause substantially caused salaries to increase in the MLB. iii)Baseball and Billions Another significant undertaking of evaluating marginal revenue product for athletes was done by Andrew S. Zimbalist in 1994. Zimbalist worked with baseball data ranging from 1986 to 1989, which is over a decade after the reserve clause was adjusted, thus players were expected to be paid closer to their MRPs. The method used here is once 16 PAGE 23 again an adjusted version of Scully's method to make it more accurate and relevant to the current time period. However, prior to introducing his method, Zimbalist criticized Scully's model for its lack of accounting for ce rtain factors that play into an athlete's value to his team. Among the factors left out of Scully's analysis are: starter's charisma, basestealers, psychological or strategic c ontributions, more importantly, any fielding contributions that the hitters would ha ve. He also goes on to note the negative contributions that an athlete may have including, "poor fielding, bad base running, contentious or selfabsorbed personal ities, and so on." (Zimbalist, 1994). One more criticism of Scully's method wa s that value was given to a player as long as he produced any kind of statistic; however, some stat istics that a player may produce could work against their team, such as a low batting average. Zimbalist sought to correct for this oversight by using statistics compared to league averages, meaning that players who are well below average are seen as negative to team's output. An additional adjustment made in this process was that of the main contributing factors to winning percent. Zimbalist didn't feel as if th e slugging average was e nough to predict win percentage; instead, he used a production parameter which combined slugging average and onbase average. He also decided to use a pitcher's earned run average, rather than strikeouts to walks ratio. Zimbalist's team revenue regression included the parameters population, income per capita (both by city), the National L eague dummy, and a multiyear trend variable. The results were significant and showed positive coefficients on such factors as population and income per capita which reflect that richer, larger market teams will usually generate more revenue. Zimbalist calculated marginal revenue product in a 17 PAGE 24 sim ilar method to Scully, by multiplying th e winning percentage coefficient by the coefficients on the performance parameters but instead took into ac count average salary which provides him with more comparative re sults. One caveat, is that their estimates, "hinge critically on the average salary per team To the extent that the average player on a team is underpaid (overpaid), our MRP estimates will be too low (too high)." (Zimbalist, 1994). A final note that Zimbalis t included is that he could only report his findings using the revenues that the teams reported, owners will often find ways to downplay and underestimate their actual earnings for reasons such as claiming losses in bargaining or saving on their own pers onal income taxes. One of the important findings in Zimbalist's publication was that the amount in which players are exploited depends largely on what stage they are in their careers. When a player is in his first two years in the league (called an apprentice), he is not eligible for arbitr ation, they do not reach eligibility for free agency until they've been in the league for si x years (between years three to six, they are journeymen). Zimbalist found that players with more than six years of experience were often paid more than their actual MRPs. Thus, depending on the length of the player's career, the amount that the owners are able to exploit players diminishes as players gain experience. iv) What's Wrong with ScullyEstimates of A Player's Marginal Revenue Product The final article that focuses on estab lishing marginal revenue products for baseball players is another reaction to the Sc ully method and attempts to compare it with the Zimbalist method mentioned previously. One of the more interest ing criticisms that this article contains early on is with Zimbalist's method, wher e he implied that those who have far less than average pr oduction factors could be imputed to have negative marginal 18 PAGE 25 revenue products. Negative m arg inal revenue products are, "not allowe d by secondorder conditions. (Krautmann, 1999) Mathematically, marginal revenue is a derivative of total revenue which has to be an upward sloping function to ma ke economic sense. Negative marginal revenue would count toward cost; thus, the idea of nega tive marginal revenue product can not exist. (Chiang & Wainwright, 2005). Another distinct ion that Krautmann chooses to make has to do with ex post versus ex ante realization of performance. This has to do with the timing of the year's statistics with the salary data. The distinction here in Krautmann's analysis is defined, where the ex ante contribution of the player is obtained by averaging his perf ormance over the previous three season and the ex post contribution is defined as the player's endofseason performance statistics." There is usually a large discrepancy in the performan ce expected from an agreed upon salary and the performance actually made by the athlete. Krautmann emphasized this distinction in his model to show the volatility in the me thod which Scully used in the establishing marginal revenue product. One of the biggest issues that most critic s have with Scully's method is that they believe his figures for marginal revenue product were usually much too high. This would explain the lack of correlations between Scu lly's estimated MRPs and the actual salaries of the players. The Krautmann method empl oys an assumption that the competitive free agent bidding process will actually line up player wages with their MRPs. Krautmann was accounting for factors such as the instin ct of a team's general management and coaching staff in evaluating the actual talent of a given player. He uses this assumption to analyze the performance of free agents' contri butions to winning. He used the results to create an interaction that applied to the players who were still held under the reserve 19 PAGE 26 clause (apprentices and j ourneymen). The interaction used created an endogenous variable called VALUE which was m easur ed with the exogenous variables for performance and overall trend. It seems to be a common thread throughout these models that the journeymen and especially the apprentices seem to be paid well below their marginal revenue products for their team s, but upon further comparison with another traditional economic model, the discrepancy could be well accounted for. This method established by Krautmann is referred to as the Free Market Return (FMR) Method (Krautmann, 1999). The human capital model is a common topi c in the subject of labor economics; it demonstrates how individuals i nvest in themselves in order to increase their wealth to society. Increasing one's human capital can invo lve difficult decisions such as choosing between college or starting a career. This ab stract analysis deviates from traditional economics as it can be difficult to view humans as a type of physi cal capital (Schultz, 1961). While most of the analyses have shown that once players reach free agency they either make or exceed their marginal revenue product, nearly all shown have that in an athlete's earlier years they fall subject to monopsonistic exploitation by the owners. Perhaps being subject to this exploitation can be looked at as another input in the human capital process, which would mean that sin ce baseball experience is applicable to the entire league, the ear lier years are providing a player wi th the necessary abilities to succeed as a veteran in the major leagues. This publication mentioned this, but never explicitly compared baseball's reserve cl ause to the human cap ital model of labor economics. 20 PAGE 27 Krautmann concludes by referencing th e discrepancy between training costs and amount that the owners save th rough exploitation using the rese rve clause. He claims that with his FMR method, the surplus that the aver age team extracts from use of the reserve clause is $4.5 million per player. He references several sources to come up with an estimation of somewhere between $36 milli on for developing players. Thus, according to the Krautmann method, teams generally cove r their training costs through usage of the reserve clause in the players' first six years in the league. Using the Scully method, the amount that owners would save with the re servation clause was about $57 Million. In Krautmann's concluding remarks he claims that his method could be fu rther applicable in the baseball labor market to reveal many other different di scriminatory practices that could be occurring at the time of his publication. v) Salary Vs. Marginal Revenue Product U nder Monopsony and Competition: The Case of Professional Basketball The majority of research and work done in establishing marginal revenue products for athletes has occurred in baseball, mostly due to its simplicity. In order to deviate from baseball, this next article focuses on establishing MRP for professional basketball players. Published by Scott, L ong, and Somppi (1985), the primary goals of this article were to determine amounts of monopsonistic exploitation in the National Basketball Association (NBA) and amounts of racial discrimination being practiced through salary discrepancies. The authors sough t to explore the differe nce in labor market structure of the NBA following a land mark federal court ruling in 1976 that effectively provided for more mobility of labor for professi onal basketball players. The terms of this 21 PAGE 28 ruling in cluded eliminating the option clause on a player's contract2 and establishing a system of "right of first refusal" for fr ee agency commencing in 1980. The right of first refusal system allowed for teams to retain the talents of their free agent players as long as they were willing to match any offer from a competing team. The authors believed that player salaries increased as a portion of th eir MRP as a result of these changes in labor market structure (Scott, Long, and Somppi, 1985). The first objective in this study was to estimate MRP. The first step was running a regression using winning percentage as the endogenous variable and multiple parameters to estimate different contributions of basket ball players. Unlike previous analyses, this one did not attempt to aggregate one posit ion's contributions by using one or two parameters. Instead, five parameters were used to represent different basketball statistics. One new aspect introduced in this process wa s for each of the five statistics recorded, there was an opposite to show how their opp onents performed. This was used as an attempt to quantify the defensive aspect of basketball by showing how effectively a player could restrict their opponent from pr oducing positive statistics. The five statistics used were: field goal percentage, free thro w shooting percentage, rebounds, assists, and fouls. For the player being meas ured, all of the parameters we re expected to have positive coefficients, except for the fouls statistic since they contribute negatively towards winning. For the opponent representations, these expectations were the exact opposite. Like in previous articles, the next part of calculating MRP was to establish a total revenue regression using various parameters expected to affect a team's revenue, while seeking out a coefficient attached to winning percentage. Other than winning percentage, the exogenous variables measured in this regression were: metropolitan population, arena 2 Similar to baseball's reserve clause 22 PAGE 29 capacity, percentage of population that was bl ack, per capita incom e, amount of years the team has been in the city, number of superstars on the team, a dummy variable showing whether the team was in playoff contention, and the percentage of players on the team who were black. The number of years a team has been in the city variable seemed interesting as it could measure a tradition aspect of sports that can be difficult to quantify. The superstar variable was difficult to quantif y as it doesn't have a clear definition. The variables measuring black population and team percentage were used to determine amounts of discrimination in the NBA. The results of the first regression all ten variables, except for home team and opponent free throw shooting percentages, were significant within the one percent level. This regression also had an R2 value of 0.71 and a statistical ly significant F value of 43.62, meaning that much of winning percenta ge was accounted for by the ten exogenous variables. This regres sion covered ten seasons of NBA data from 19701980. The analysis went on to estimate the revenue e quation which generated st atistically significant coefficients for the sought af ter variables. The coefficient for winning percentage was statistically significant within the five percen t level. Also, the coefficients measuring the discrimination variables were statistically significant within the one percent level. These results allowed the study to calcul ate marginal revenue product for salaries measured in the 198081 season, which was the first season using the major changes in free agency. Representative salaries were cal culated for samples of players still on long term contracts from prior to the rule change and salaries for play ers who became free agents as a result of the rule changes. Th ere was significant evidence that exploitation had occurred prior to the free agency changes as the players' salaries were on average 23 PAGE 30 much sm aller than the players who were freely able to sell their labor for an economically competitive price. The final calculation of this study was to determine if racial discrimination was practiced by team owners. A regression was run using salary as the endogenous variable and MRP and a race dummy variable as the exogenous variables. The race variable was not statistically significan t, which lead to the conclusion that racial discrimination did not occur considerably in the NBA at this time. The concluding remarks of this publication maintained that NBA salaries would begin to approach MRP overall and racial discrimination did not signifi cantly affect revenues or salaries at this time. vi) Pay, Performance, and Competitive Balance in the National Hockey League The final article to be revi ewed that attempts to establish a marginal revenue product for a league of professional athletes will be pertain to professional hockey players. Richardson establishes marginal re venue product for hockey players to attempt to show, "the effects of free agency, definitio ns and measurement of competitive balance, and the effect of the entry draft on compe titive balance" (Richardson, 2000). Richardson's equation to estimate different aspects of team revenue is similar to the Zimbalist method as he employs a lag for winning percent to include data from both the 198990 season and the 199091 season. An additional factor in Ri chardson's analysis is the inclusion of playoff games. A team is able to make more revenue from a playoff game than an average game, because these games are more important drawing more fans. A team is able to raise ticket prices which increases ga te revenue, since fans have a much higher demand for these games. Like the Zimbalist method, Richardson includes a trend variable to show the salary trend ove r the years 198990 to 199596. On e more parameter that was 24 PAGE 31 added was a dummy whi ch stands for the lo ckout year in the 199495 season. Richardson avoids using fixed effects data, which pertains to data that generally does not change over time; such as team location, size of their arena, or per capita income. All of Richardson's coefficients were significant within the five percent level. The lockout variable quantifies the amount th at the owners lost to be $9.409 million. Richardson established his marginal revenue function which provided a coefficient in front of winning percentage to be establ ished as the interaction between winning percentage and marginal revenue. He then ran a regression using winning percentage as the endogenous variable and the various player statistics as the exogenous variables. This winning percentage regression had several inde pendent variables due to hockey's various positions. He used goals per game and assists per game as the forwards' contributions. For defenders, Richardson used points3 per game and penalty minutes per game. For goalies, he used save percentage. Once again, all of his coefficients proved to be significant to the five percent level, and in both equations all of the coefficients had their expected signs. Next, to establish marginal revenue product, he multiplies his results for marginal revenue by change in winning percentage between each player. Richards on also attempts to take his results and apply them to the players on the roster who ra rely play by dividing his data by number of games played. One problem with this is that when reserve players do get to play in a game they do not play very long. Therefore, without data showi ng amount of minutes played to divide the output parameters by, it is likely that the va lues for the reserve player's MRPs would be undervalued. A goal scor ed with only five minutes of play time should be worth more than goal scored with tw enty minutes of play time. This is not an 3 Points equal Goals added to Assists 25 PAGE 32 issue with the roster players, since they us ually receive nearly e qual amounts of playing tim e. Richardson's next undertaking was to establish whether player s are underpaid or overpaid using a marginal salary variable that is a player's salary w ith the average salary of the reserve players at that position subtract ed from it. He then subtracts this marginal salary variable from the marginal reve nue product, which gives a surplus factor. Richardson's results showed that each position had a couple million dollars worth of surplus across the league, which lead him to co nclude that there was not a large degree of monopsonistic exploitation in the NHL. Richar dson also made assertions of the human capital model and tested this by regressing the surplus factor over time to see if it diminished. The results only proved to be significant with rega rd to forwards. vii) Measuring Managerial Effic iency: The Case of Baseball The final article in this literature review employs an analysis that has been a side note in many of the previous articles disc ussed, and was published by Philip Porter and Gerald Scully in 1982. Managerial efficiency has been overlooked so far as it is difficult to quantify and could potentially throw off results when not assumed away. To quote this article, "The managers of major league baseball teams are the economic agents responsible for transforming inputs into percen t wins." Porter and Scully use the Cobb Douglas isoquant relation to help measure efficiency with the inputs of hitting and pitching (Porter and Scully 1982). Scully follows his original method by using slugging average to represent hitting and the team strike out to walk ratio to re present pitching. These were divided by the league averages for their va rious measures to show relative team slugging and pitching 26 PAGE 33 averages. T hrough this, a chart was developed comparing the mean managerial efficiency of many of the various managers that coach ed between the years 19611980. Their results showed a correlation between efficiency and durability implying that managerial efficiency increases as they accumulate mo re experience. The empirical results were: "with efficiency at about 89 percent of poten tial, managerial performance increments at 0.8 percent per year at a diminishing rate reaching a maximum of 94.4 percent after 12.5 years" (Porter and Scully 1982). Their results al so suggest that there is a realistic decrease in efficiency among managers th at change teams more often. Porter and Scully go on to measure th e marginal revenue product of managers using their calculations of ma nagerial efficiency and the coefficient relating a percentage increase in team winning percent to revenue. Th e results of this method did not appear to be accurate since they showed that managers have similar marginal revenue products as star players. The results for managerial efficiency appeared to have accurate methodologies, because it was proven that the most regard ed managers in the MLB had the highest calculated efficiencies. An establishment fo r the MRP of managers was not sufficiently completed since the calculated MRPs we re much higher than any team would realistically pay. Looking ahead to the analysis to be conducted in this work, many of the methods and calculations to be performed were drawn from the work produced by these founders who originally esta blished MRP in the industry of professional sports. 27 PAGE 34 Chapter 2: Methods The experimental procedure of this research uses the empirical tools of econometrics. Econometrics "consists of the a pplication of mathema tical statistics to economic data to lend empirical support to the models constructed by mathematical economics and to obtain numerical results," (Tin ter, 1968). The analys is of mathematical statistics often uses experimental data; da ta that gets obtained by conducting a specific test. Econometrics commonly uses observationa l data collected from prior events not pertaining to the model directly (Spanos, 1999). The data collected was observed from players and teams in the NFL and was used to calculate relations hips between onfield performances, winning, and team revenues. Hence, the data used in this analysis is observational. It consists of statistics from the 2008 NFL season. The first step in deriving the formulat ed model was to break down the analysis into different components that compose regr essions. The analysis sought to establish marginal revenue product to estimate the amount of monopsonistic exploitation in 2008. Football has many specialized positions and each one lends different type s of statistics to reflect the performance of an athlete. In addition, the data collected for offensive and defensive positions possesses different parameters; thus each position was measured using a different regression and data set. Regression Analysis This process relies on the use of regr ession analysis. "Reg ression analysis is concerned with the study of th e dependence of one variable, the dependent variable, on one or more other variables, the explanatory variables with a view to estimating and/or predicting the mean or average value of the fo rmer in terms of the known or fixed values 28 PAGE 35 of the latter." (Gujarat i, 2003). A regression analysis is calculated using m any different techniques depending on the type or number of endogenous and exogenous variables. This analysis differs from those discussed ea rlier in that each pos ition's regression was split up due to restrictions found in their da ta collection. Regressions showing various athletes' contributions towards winning were calculated for each position. Quarterback Performance Regression The first regression was the quarterback binary regression which measured a quarterback's passer rating's effect on winning. The endogenous va riable in this regression was game outcome which is not quantitative by nature. Therefore, it is measured using a dummy variable. Dummy vari ables are used to quantify two different outcomes for an entry and uses either one or zero to record the observation. A value of one would mean participation and zero woul d be the contrary. A regression using labor force participation as the endogenous variab le could be run with exogenous variables such as unemployment rate or family income (Gujarati, 2003). In econometrics, binary regressions are not as common as regressions with continuous quantitative endogenous variables. Binary regression techniques are cal culated differently than the typical method ordinary least squares. The first regressi on is computed using the probit model. Probit Model The probit or normit model uses the endoge nous variable, titled the logit, as the probability of a certain outco me occurring. These regressions can have domains from negative infinity to positive infinity but ma y only have ranges from zero to one. The probability is derived from inputting the exogenous variables in a particular model. The probit model is similar to the logit model which uses the natural log of the probabilities of 29 PAGE 36 the exogenous variables to dete rm ine a number that represents a probability between zero and one. The primary difference between the probit and logit models is that the probit model assumes a normal distribution while th e logit model assumes a similar logistic distribution (Gujarati, 2003). Th e probabilities of the exogenous variables, also called the regressors, are calculated using the expected probability value of the endogenous variable for a certain input of each exogenous variab le. This regression equation computes coefficient values for the exogenous variables which show the direc tion and magnitude of their relation to the endogenous variable. (Kramer, 1991). The results will also use a term called the McFadden R2 which deciphers the amount that the endogenous variable is explained by the parameters. Additionally, a probability value for each variable is used in the results to determine if they are statistica lly significant. Statistical significance is the overall goal for an analysis such as this because the probability value defines the likelihood of the results. If the results are not shown to have a high probability then they are incomplete in explaining the relationships sought after in the regression. Revisiting the Quarterback Performance Regression Now that the method for modeling this regr ession has been defined, the different variables that the regression contains and why they are included can be explained. As previously noted, the endogenous variable in this circumstance is the dummy variable detailing the games outcome, with quarterb ack rating being the onl y exogenous variable. Thus, Equation 21 appears as follows: QBW_L = c0 + c1QBRATE Equation 21 QBW_L represents the exogenous du mmy variable produced by each performance in the data set. The winloss labe l (W_L) is preceded by QB as it represents 30 PAGE 37 the re sults from the specific quarterback data set.4 C0 represents the constant coefficient, which covers the interactions of everything not explaine d by the exogenous variable. QBRATE represents quarterback rating, and its coefficient c1, shows the effect that quarterback rating has on the games outcome.5 The constant coefficient was expected to be large as was the McFadden R2 term.. The expectation for the QBRATE parameter was that its coefficient would be positive and sma ll in comparison to the constant coefficient. The next step is to look at the rating system that evaluates a quarter back's performance. Quarterback Rating On average, the quarterback is the highest paid player on a team. He is known as the "field general" and is regarded as the t eams offensive leader. The expectation for the QBRATE coefficient was higher than the coe fficients for the runni ng back rating and wide receiver parameters. A quarterback's ra ting is determined from a mathematical formula derived using four different input factors. The fo rmula was developed in 1971 by an executive at the National Football League Hall of Fame named Don Smith. Prior to this, there was no method for ranking quarterb acks that wasnt comparative by nature. Every previous ranking that the league had used was dependent on the statistics of a quarterbacks peers (Steinberg, 2001). As a result, in 1971, league commissioner Pete Rozelle sought out his best statisticians to determine a new method to rank quarterback performance that could be computed independently. Originally Smith wanted his system to be based on a zero to one hundred scale. Instead of using averages to provide ranks for his data, he used ideal performances at the time to generate his standards. The four inputs to passer rating are co mpletion percentage, 4 The labels in front of the endogenous variables for winning will subsequently refer to the position to which the data set corresponded. 5 It is necessary to note that the co efficients produced by logit equations do not represent marginal effects. 31 PAGE 38 yards per passing attempt, percentage of thro ws intercepted, and percentage of throws resulting in touchdowns. If a quarterb ack has average numbers, then each factor would be equal to around one. If the quarterback is we ll above average, they earn scores of approximately two. A perfect passer rating woul d score 2.375 in all four categories. Since Smith desired his system to be scaled from zero to one hundr ed, he multiplied the results by a hundred and divided by six, making th em equal to two thirds. However the maximum score on any statistic was 2.375, so th e maximum QB rating came out to be an arbitrary 158.3 (Steinberg, 2001). The league incorporated the QB rating in 1973 and still uses it as the primary quarterback ranking determinant. The cal culation of the QB rating overlooks many factors that make a quarterback ultimately successful. It om its intangible factors such as leadership and ability to perform in high pressure situations. Furthermore, tangible factors such as running ability or wide r eceiver drops are excluded. QB rating is not perfect but does a successful job at evaluating a passer, which is the primary role of a quarterback. Running back Performance Regression The NFL does not employ a standard sta tistic for running backs as it does for quarterbacks. Previous aut hors used a single parameter such as baseball's slugging average to evaluate a hitters performance. This method overlooked additional roles of these players, such as defense, that are not included in the para meter. Another option would be to include multiple parameters to evaluate performance, but this increases the probability that multicollineari ty can occur. This analysis employs its own methodology to determine the overall rating of a running back in a game. The method developed uses 32 PAGE 39 com parative data to evaluate the most efficien t performances in the sample data set. The process will be similar to the method used by De Oliveira and Callum in their article titled, Who's the Best? Data Envelopment Analys is and Ranking Players in the National Football League The cited article utilizes Data Envelopment Analysis which ranks "decisionmaking units" (DMUs) based on various input and output measures. (DeOliveira and Callum, 2004). The authors ap plied this method to both quarterbacks and running backs. The running back input measur es were rushing attempts and receptions. The output measures were rushing yards, rece iving yards, total t ouchdowns, and carries per fumble. Different weights were applied to the input an d output measures to estimate the DMUs efficiency. A similar method was adopted in this pr ocess to empirically define a ranking measure for the performance of running backs. The only input measure used to calculate a running back's performance was number of ru shing attempts as rushing is the primary role of a running back. The output measures designed to evaluate running backs were yards, touchdowns, and yards per fumble. Th e yards per fumble evaluation was lagged by one in order to avoid division by zero. The weights of each variable were calculated by taking the reciprocal of the av erage of each output measure used across the data set. For example, if running backs accrue on averag e 50 yards per touchdown scored, then a touchdowns weight would be approximately 50 times the weight of a yard gained. The results were interpreted as the running back's gross rating. The net rating was calculated by dividing the gross score by the input meas ure multiplied by its weight. Input variable weights were calculated in same way. The net score was used to rank each individual 33 PAGE 40 observation and became the RBRAT variable used in the running back performance regression (Equation 22). RBW_L = c0 + c1RBRAT Equation 22 Once again, a binary probit model was used to calculate the regression in Equation 22. In Equation 22, c0 represents the constant coefficient and c1 represents the coefficient to relate RBRAT to game outcome. The expectation for this coefficient was that it would be small and positive, and th at this regression would produce a smaller McFadden R2 than the quarterback performance regression. Wide Receiver Performance Regression The regression relating wide receiver performance was calculated in a similar method as the running back performance regre ssion using a variable designed within this analysis to evaluate overall wide receiver performance. The wide receiver rating, titled WRSCORE, was calculated in a comparable method as the running back rating except with different parameters. Only weighted output measures were used to aggregate the performance of a wide receiver ranking each across the sample data. The output measures used were receptions, yards, touchdowns, and receptions without a fumble lost. The receptions without a fumble lost measure wa s once again lagged by one to avoid division by zero. The wide receiver score generated by these calculations was used to evaluate their performance in terms of winning in Equation 23. WRW_L = c0 + c1WRSCORE Equation 23 The coefficient c0 represents this regression's constant coefficient while c1 was the sought after performance coe fficient. The overall rating scor es used by the first three regressions to evaluate player performan ce were designed to avoid multicollinearity. 34 PAGE 41 Multicollin earity Multicollinearity is a common issue with regressions containing multiple explanatory or independent variables. It occurs when different independent variables boast a correlation with one anot her. As a result, the computed coefficients possess very large standard errors distorting the accuracy of the measurement. The primary affect of multicollinearity is that coeffi cient estimates will not have small standard errors; however, the estimates will remain unbiased (Gujarati, 2003). The variables used in the performan ce regressions were employed to limit the amount of this phenomenon which could easily occur due to "constraints on the model or in the population being sampled" (Gujarati, 2003). These constraints occur with each positions output parameters as they are commonly highly correlated with one another. For example, wide receivers aren't able to generate yards without making a reception. Quarterbacks must complete passes in orde r to throw for a t ouchdown. All of these output statistics interact with each other. Aggregating the pe rformances for all positions into one variable seemed to be the best met hod to avoid issues with collinearity. It also accomplished an unintended goal of determining performance ratings for two positions that did not have a readily record ed value prior to this method. Defensive Linemen Performance Regression The fourth regression in this analysis was th e first step in the defensive side of the calculations and models the amount that defens ive linemen contribute to a team's winning percent. The primary difference between the defensive and offensive estimations was that the defensive estimations were based on data from the entire 2008 season, rather than data given from each individual game played. Therefore, our endogenous variable to be 35 PAGE 42 determ ined in this regression will be team winning percentage. Ordinary least squares models were used to estimate the remaining regressions. Ordinary Least Squares The Ordinary Least Squares (OLS) met hod for multiple regression interpretation is one of econometrics' most common proces ses. The OLS model computes a continuous endogenous variable rather than the binary va riable from the probit model. Ordinary refers to the exogenous variab les being linear. The OLS me thod is computed using the minimum of the parameters residual sum of squares (Gujarati, 2003). To clarify, the equations that make the regression form a line in space. The different observations all form different illustrations of the line. The OLS method seeks to fit them to an aggregate equation by minimizing the values of the figure s error terms that vary from the best fit line of regression. The error te rm is the amount that an obs ervation varies from the regression line of best fit. Regression lines are made from aggregating the observations with the most limited error terms. OLS assumes homoscedasticity, meaning that error terms have constant variance. This assumpti on is incorrect when he teroscedacity occurs, because variances of error terms are infreque nt so significance test estimations may be imprecise (White, 1980). Part of the OLS models output is the equation for the line of best fit for the entire regression. The model produces an R2 value which provides insight into the behavior of the function. This form of R2 value is different than the McFadden version that was produced from the binary regression. The new R2 term is still defined as a measure for "the goodness of fit of the regression equa tion; that is, it gi ves the proportion or percentage of the total vari ation in the dependent variab le" (Gujarati, 2003). The R2 value 36 PAGE 43 is im portant to the results b ecause it demonstrates the amount that the model explains the relationship between the exogenous and endogenous variables. The analysis is split up between offensive and defensive regressions, so the R2 value was not expected to be close to one. Although, Scully's estimates show high R2 values despite pitchers being the only position having their defensive abilities accounted. Win Probability Added (WPA) The exogenous variable measuring the performance for all defensive positions measured was their Win Probabil ity Added (WPA). Defensive performance indicators are difficult to procure because they are often misleading. For example, opposing offenses are allowed more plays ag ainst an inferior defense meaning that tackles could be a misleading statistic as players on inferior teams have more opportunities. WPA uses score, time, down, distance, and field position to estimate how likely each team will go on to win the game (Burke, 2010). WPA quantifies the results of any single play and calculates the change in the probability of winning the game from that point. This statistic can calculate the most dramatic plays in any NFL game over a time period. WPA was used in the regression du e to its relevancy despite it not directly measuring the true abil ity of the player. WPA is used to measure the result of a part icular play, but it ca n still be applied to athlete's contributions towards winning. Summing the WPA of plays that an athlete contributed can quantify their ability to produce results when it matters most. These contributions are more effective at evaluating winning ability than the primary parameters, since intangibles such as ability under pressure are included. The rankings for WPA were analyzed to check the legitimacy of these borrowed statistics. The results 37 PAGE 44 turned out to be a straightforward ranking of the greatest players in the league at their respective p ositions. This simple test makes thes e statistics veritable enough to be used in calculating an athlete's marginal revenue produ ct through their ability to help the team win when it matters most. Defensive Linemen Performance Revisited Using these parameters, the contributions of defensive linemen were computed using team winning percentage as the endogenous variable. Equation 24 is the defensive linemen performance regression. DLWPCT = c0 + c1DLWPA Equation 24 The DLWPCT variable represents the winning percentage of each player's team in the data set. DLWPA demonstrates contribu tions of both defensive ends and defensive tackles towards win probability. DLWPA's coefficient c1 was expected to be positive possessing the second highest R2 value out of the defensive positions, because their contributions are likely more difficult to quantify than lin ebackers, but are involved in plays quicker than the secondary. Linebacker Performance Regression The performance of linebackers was cal culated in an identical method as defensive linemen using the WPA statistic as the parameter being related to winning percentage. Equation 25 repr esents this regression. LBWPCT = c0 + c1LBWPA Equation 25 The expectation for c1, the LBWPA coefficient, was that it would be nearly the same as the DLWPA coefficient. The R2 term from this regression was expected to be the 38 PAGE 45 larges t out of the defensive positions, becau se linebackers are the central unit of the defense and have the easiest contributions to quantify. Secondary Performance Regression The secondary position representing players in the defensive backfield was calculated using WPA to eval uate performance. The sec ondary positions consist of cornerbacks, strong safeties, weak safeties, a nd other variants of these positions such as nickelback. The players that play in the defensive backfield are mostly used in defending against a pass, although safetie s often contribute on running pl ays. The contributions of players in the secondary are the hardest to quantify out of any pos ition. The role of a cornerback is to guard a receiver that he is assigned to in mantoman coverage, or an area of the field assigned to him in zone co verage. If the ball is not thrown to their receiver or their zone, then that player gene rally has won that play. If a cornerback does not have any interceptions or pass deflec tions it could be because quarterbacks are hesitant to throw the ball in their directi on. Thus, quantifying their contributions can be very misleading. Safeties represent the last line of defense before the end zone, so while their contributions are positive for the team, it usually means that the defense has broken down in some way. Equation 26 sought to qu antify these contributions' effects towards winning in an OLS regression. SWPCT = c0 + c1SWPA Equation 26 The perceptions regarding th e secondary position lead to smaller expectations for this regression. Even though ther e are usually more secondar y players on the field than the other positions, the lack of easily calculated contributions lead to an expectation of 39 PAGE 46 the sm allest R2 value out of the defensive players measured. The coefficient for SWPA was expected to be a very small positive number between zero and one. Regressions Relating Winning Percentage to Revenue The next step of the analysis was to apply a monetary value to winning based on revenue. To accomplish this, an OLS regression was run with the endogenous variable team revenue. The exogenous variables for this regression were winning percentage and the player's city's gross domestic product (GDP) per capita. The goal of this regression was to establish a statistically significant coe fficient for team win percentage to be used in calculating marginal revenue product for an athlete's contributions. GDP per capita was included in this regression as an attempt to increase the R2 value because it is the most widely accepted standard of living m easurement. The expectation for GDP per capita was positive and small due to additiona l factors that affect team revenue. The reasoning behind the positive coefficient expectation was that teams that reside in cities with a higher standard of living should be ab le to sell more tickets and merchandise. The expectation for the win percentage coefficient was positive with a value that represents a portion of the team's total overall revenue. This expectation was determined because winning should draw more fans to watch games, increasing team's gate revenue as well as other sources of revenue, such as concessions and merchandise Equation 27 is the first attempt to relate winning percentage to reve nue using the defensive linemen data set. DLREV = c0 + c1DLWPCT + c2DLPCGDP Equation 27 In the regression, DLREV represents team revenue for the defensive linemen data set. DLWPCT refers to team's winning per centage, and DLPCGDP represents the team's city's per capita GDP. 40 PAGE 47 The coefficient relating winning percentage to team revenue should be the same regardless of position. It wasn't necessary to run this regression for all six different positions; however, the data was recorded by position separately, therefore the regressions are labeled by position. The most im portant part was to create a rational value to multiply the performance coefficients with to create MRP evaluations. The first regression of this type was calculated in the defensive li neman data set and the second used the linebacker data set to confirm the coefficient. The Equation 28 represents the regression run to confirm the results from Equation 27. LBREV = c0 + c1LBWPCT + c2LBPCGDP Equation 28 The variables in Equation 28 are the same as Equation 27; however, they are preceded by LB instead of DL because they correspond to the linebacker data set. Once completed, this regression confirmed the coeffici ents to be used in calculating marginal revenue product for the six different positions measured. Relating Gate Revenue to Winning The previous section consisted of re lating winning to overall team revenue; however, it is widely accepted that gate re venue could be a better measurement of a team's ability to contribute towards revenue through winning. This idea holds more ground in the NFL than in any other major sports league, due to the large amount of revenue sharing in the league. Revenues receiv ed from the NFL's lucrative television and radio broadcasting deals are split evenly among the 32 franchises (Moorhead 2006). Local revenue consisting of gate, concessi on, parking, and local TV revenues provide incentive for teams to win. The majority of local revenues are kept by each team; however, 40% of gate revenues are allocated into a pot to be split equally among all 41 PAGE 48 team s (Fort, 2006). The coefficients for gate re venue were calculated in the same way as standard revenue with the only difference being that gate revenue is the endogenous variable. This part of the analysis maintain ed win percentage and GDP per capita as the parameters and used the defensive linemen da ta set first with the linebacker data set confirming the results. Equation 29 was the re gression run to relate winning percentage with team gate revenue. DLGREV = c0 + c1DLWPCT + c2DLPCGDP Equation 29 The expectations were the same as the standard revenue regressions, except the coefficient using gate revenue would only be a portion of the coefficient produced using standard revenue. Equation 210 was the c onfirmation regression run for Equation 29 using the linebacker data instead of the defensive linemen data. LBGREV = c0 + c1LBWPCT + c2LBPCGDP Equation 210 The coefficients generated by these two e quations were used to calculate marginal revenue product for gate revenue6 and also see the average percentage that gate revenue contributes towards total revenue. Calculating Marginal Revenue Product The calculation of marginal revenue pr oduct for each offensive performance and each defensive player involves multiplying the value of the variable measured in the performance regressions by the coefficient produced by each position's performance regression. This value was then multiplied by the coefficient produced in the regressions showing the effect that winning has on team revenue. For example, to calculate quarterback MRP, each quarterback's passer rating was multiplied by c1 from Equation 21. This value was then multiplied by c1 from either Equation 27. The product of this 6 To be referred henceforth as Gate Marginal Revenue Product or GMRP. 42 PAGE 49 multiplica tion represents the MRP for quarterbacks. Equation 211 is the formula used for calculating quarterback MRP. MRP (QB) = QB Rating x c1 (From Equation 21) x c1 (From Equation 27) Equation 211 For each position, the first term would be th e rating determined to evaluate player performance, the second term would be th e coefficient correspondi ng to that position's performance regression, and the third term remained constant for all six positions. Gate MRP was calculated in the exact same way, ex cept instead of using the coefficient from Equation 27, the coefficient from Equation 29 was used to calculate athlete's gate MRP. Once both types of MRP values were calculat ed, the results were compared to actual salaries of the players measured to determ ine rates of monopsonist ic exploitation by position. Calculating Monopsonistic Exploitation The method for determining monopsonist ic exploitation was duplicated from Scully's method by subtracting actual salary from calculate d marginal revenue product, then dividing this difference by marginal re venue product. These calculations are shown in Equation 212. Monopsonistic (MRP Salary) Equation 212 Exploitation MRP Instead of using every measured athlete' s salary from 2008, representative salaries were recorded for every posit ion besides quarterbacks. A sample of salaries were recorded at the minimum, first quartile, me dian, third quartile and maximum values in order to aggregate the large volume of data. Provided MRP was larger than salary, this 43 PAGE 50 calculation would generate a decim al between zero and one corresponding to the rate of monopsonistic exploitation. A rate equal to one would represent perfect exploitation while a rate of zero would mean that no expl oitation occurred and the labor market was perfectly competitive (Scully, 1974). If the rate was a negative number, then reverse exploitation occurred meaning that player s had more bargaining power. The rate of monopsonistic exploitation for ga te revenue was calculated in the same way except using the GMRP instead of MRP. 7 7 The same recorded salaries were used to determine gate monopsonistic exploitation. 44 PAGE 51 Chapter 3: Results Quarterback Perfo rmance The first step in formulating a marg inal revenue product for players was to determine their contributions towards winni ng using onfield pr oduction ratings. The maximum possible score for a QB rating is 158.3 which was achieved by a player only twice in 435 games measured, both resulting in victories for their teams. Seneca Wallace had the highest QB rating (128.9) that didn' t achieve a victory. The average QB rating was approximately 87 with a median of 85.3. The QB rating data set was relatively normally distributed and had minor skewness. The data set had a JarqueBera value of less than one, meaning the data was nearly normally distributed. The minimum value of any quarterback rating measured was 12.3, pr oduced by Jake Delhomme, which resulted in a win despite his four interceptions. The defense made up for Delhomme's performance by holding their opponents to si x points and the running back, DeAngelo Williams, contributed by rushing for 140 yards. Quarterbacks were shown to have the most dramatic effect on their team's ability to win because Equation 21 had a much higher R2 value than the other position's performance regressions. A binary probit regression was used to calculate the results for the performance coefficient achieved. The quarterback rating variable had a positive coefficient of 0.02 with a McFadden R2 term of 0.18. This is a very high R2 term considering that a quarterback is just one position out of many, and accounts for nearly 20% of the game's outcome. The coefficients obtained in the regressions relating player performance to game outcome or winning percentage are not to be explicitly interpreted, as they are initial 45 PAGE 52 steps in a la rger process. The likelihoodratio (L R) statistic is used to determine the fit of the models being measured. It produced a value of approximately 113.0 making the probability value 0.00, defining its statistical si gnificance. Thus the regression as a whole can be considered significant, making it applic able as the first step in the quarterback analysis. Running Back Performance The running back data consisted of 689 observations of both running back and full back performances. The highest runni ng back rating was performed by Denver's Tatum Bell who had 10.8 yards per carry, no fumbles, and two touchdowns. Bell's rating for this game was 10.86, which is well above the mean score of 3.0. The median score was 2.27 and the lowest score calculated wa s approximately 0.84. The lowest score was performed by Ryan Grant for the Green Bay Packers. Grant had 15 carries for 20 yards, with no touchdowns, and one lost fumble, in a loss to the Tampa Bay Buccaneers. The running back data set was skewed to the ri ght at a rate of 1.57 and had a very high JarqueBera value of approximately 534 crea ting a probability of 0.00 meaning the data varied greatly from a normal distribution. A binary probit model was used to cal culate Equation 22. The model produced an LR statistic of approxim ately 36 making the model's pr obability value 0.00 achieving statistical significance. The McFadden R2 term in this model was 0.038, showing that running backs have about onesixth of an e ffect on winning compared to quarterbacks. Many NFL teams employ the two running back sy stem. This occurs when the two best running backs on the roster receive nearly equal carries. The RB rating variable is measured for each individual performance, cu tting the contributions of each running back 46 PAGE 53 in half. The perform ance coefficient for running backs was approximately 0.18 with a tstatistic of 5.8 making its probability 0.00. Thus, the coefficient for running backs achieved statistical significance and can be used to confidently produce a marginal revenue product value for each running backs performance. Wide Receiver Performance The variable used to create a coefficien t relating wide receiver performance to winning was WRSCORE, which was intuitively cr eated in this work using various output measures produced by a wide receiver through the course of a game. The highest score produced in 1,092 performances was achieved by Randy Moss who played for the New England Patriots. He had 8 receptions for 125 yards and 3 touchdowns which helped his team earn a 4828 victory. The score generate d from his performance was approximately 14.7, which is well above the middle scores. Th e mean score in the data was 4.0 with a median of approximately 3.37. The wide receiv er data was highly skewed to the right measured at 1.03. This resulted in a high Jar queBera value meaning that the distribution deviated substantially from normal. The minimum score in the data set was a 0.6 produced by Will Franklin who played a minor role in the Chiefs victory over the Denver Broncos. This is supported by the small coefficient of 0.06 produced in Equation 23. Unlike quarterbacks, or even running backs, there are approximately 35 wide receivers that have play time in an NFL game. Thus each individual receiver contributes only a fraction of the total contributi on from their position. This is supported in the results as secondary and lower receivers have smaller ma rginal revenue products than the starters. Statistical significance was achieved in Equation 23. The coefficient relating WR Rating to wining was approximately 0.06. The pr obability value show ing the statistical 47 PAGE 54 significance of the coefficient was within the desired 0.05 area, hence the results achieved in this reg ression are appropriate to the use in the next part of the procedure. Equation 23 had a small McFadden R2 term compared to the qua rterback results with a value of approximately 0.01. Once again, the LR test had a probability value of 0.00, proving significance for the entire regression. Defensive Lineman The data used to calculat e the marginal revenue products for defensive players varied greatly from offensive players. Defensive player's co ntributions are more difficult to quantify. The accurate evaluation of de fensive performance was obtained after adjustments. The use of the win percentage added (WPA) statistic wa s critical because it was generated by statisticians analyzing game film to evaluate the performance of an individual player after each play. The group who measures WPA doesn't publish their results for each individual game, so the defe nsive performances measured are aggregated across the entire season (AdvancedNFLstats.c om, 2008). Hence, winning percentage was the endogenous variable to be correlated w ith WPA. Winning percentage is a continuous variable, therefore the method used for mode ling defensive performances was ordinary least squares (OLS) including the White test for heteroskedasticity. The defensive linemen data set include d 128 observations a nd contained both the defensive end and defensive tackle positions WPA was the best statistic possible to evaluate the performances of these players since their contributions often are unnoticed. Defensive end's contributions are quantified ea sier than defensive tackles as they often set the edge on run plays, allowing them to make tackles more often. Additionally, defensive ends have more success on passing plays since they can get around the edge of 48 PAGE 55 the line to rush the quarterback which is easier than going through th e middle of the line. These assertions are supported by the data se t as defensive ends are ranked m uch higher on average. The top two performers in the data se t were Dwight Freeney and Robert Mathis who had WPAs of 1.81 and 1.75, respectively. Both Freeney and Mathis play on the Indianapolis Colts who had the second highest winning percentage in the 2008 regular season. The mean of the defensive linemen da ta was approximately 0.81 with a median of 0.715. The lowest WPA out of the group was 0.25, which was performed by Josh Thomas who played for the Colts. The defensive linemen data was skewed to the right at a rate of 0.71 with a JarqueBera level of approximately 11.6. Mathis' and Freeney's performances were outstanding to the point that they could almo st be considered outliers. The Equation 24's results were succe ssful in producing a coefficient to correspond with a defensive lineman's im pact on winning based on his WPA. The coefficient produced was approximatel y 0.11 with a probability of 0.0123. The probability value confirms that our coefficient is statistically significant. This Equation 24 was calculated using the OLS method due to the continuous dependent variable. OLS was used instead of a binary re gression, therefore the McFadden R2 value is absent. Instead, a normal R2 value was calculated to be 0.03. Additionally, OLS was used instead of probit, which generated an Fstatistic instead of an LR st atistic. The Fstatistic is the standard measure to evaluate the ability to a ccept or reject the results of the model. The Fstatistic was approximately 4.88 which achie ved a probability of 0.029 lying within the =0.05 confidence interval allowing the model to be accepted overall. 49 PAGE 56 Linebackers As quarterbacks are generally regarded as the offensive leaders of a team, linebackers are often the defensive leaders. Th ey play between the defensive line and the defensive backfield and have substantial c ontributions in both running and passing. There are two types of linebackers: outside and in side. The middle linebacker is often the defensive captain since they are the core of the defensive unit. In the linebacker data set, the highest WPA was 2.3, which is the most out of any defensive player in any position. This was performed by Jonathan Vilma of th e New Orleans Saints, who only won half of their games in 2008. The Saint's second best lin ebacker that year wa s Scott Fujita who's WPA of 0.93 was the median of the data set. The mean WPA for linebackers was 0.95. The NFL Defensive Player of the Year Award was awarded to James Harrison who was an outside linebacker on the Pittsburgh Steelers. Harrison's WPA for the season was 2.14 which ranks third among any defensive player measured, however if post season data was included Harrison would have been at the t op of the linebacker rankings for WPA. The minimum value for linebacker WPA was 0.13 pe rformed by D. D. Lewis on the Seattle Seahawks. The data set was slightly skewed right at a level of 0.55, thus if a couple outliers were thrown out, the data would be nearly normally distributed since only 3 players had WPAs greater than 2. The JarqueBera was the second lowest out of any position with a valuation of approximately 6.7 having a statistically significant probability of 0.035. Equation 25 also produced statistically significant results. The Fstatistic was nearly 11 generating a probability of 0.001 making the model acceptable overall. Linebackers achieved the highest R2 value out of any defensiv e position valued at 0.08 50 PAGE 57 supporting the idea that they' re the most infl uential players on defense. The coefficient that this regression produced for lineback er WPA was 0.122 and had a tstatistic of 3.56, generating a probability value of 0.00. This falls within the =0.01 confidence interval making it statistically significant and able to be used in the process of calculating linebacker's marginal revenue product. Secondary Secondary players usually have more men on the field than linebackers or defensive lineman, thus more lines of data recorded for this category than the other defensive positions. There were 180 observations measuring the WPA produced by cornerbacks and safeties. The best rating produced among these players was a WPA of 2.02, performed by Will Allen of the Miami Dolphins. When examining the secondary positions' statistics there is a mean of a pproximately 0.75 with a median of 0.67. The distribution of the data shows some skewness to the right due to ou tliers, evaluated at 0.675. Will Allen's WPA was far above the seco nd ranked defensive back Chris Gamble's WPA score of 1.74. The minimum score out of the secondary players was 0.10 by Usama Young of the New Orleans Saints. The JarqueBer a for this distribution was slightly high, with a value of approximately 14 and a pr obability of 0.00 which makes it significant. Unfortunately, Equation 26 was the only position's regression that did not produce significant results. The re gression had a very low Fstat istic of 1.5 resulting in a probability of 0.22. The coefficient produced in this regression was approximately 0.05 with a standard error of 0.04 and a low tst atistic of 1.3. This re sulted in a probability value of 0.19 which would only be significant in the =0.2 interval. The R2 generated in this model was extremely small at 0.008, meaning either secondary players are not 51 PAGE 58 im portant towards the outcome of a game, or th ere is a problem in determining the results. Since the other R2 values weren't close to being that small, it would appear that there is either an error in the data set or the calculations. While these results weren't significant, the marginal revenue product for secondary players was still calculated, but any conclusions generated from these results could not be backed up econometrically. One possible explanation of the nonsignificant resu lts could that secondary evaluations are often misleading when quantified. Relating Revenue to Winning Equation 27 was significant overall with a high Fstatistic of 12.8 generating a probability of 0.00. The R2 value in this regression was 0.17, meaning that many factors still exist that explain where revenue is ge nerated outside of winning and per capita city income. Originally, metropolitan population wa s considered in this regression, but its calculated coefficient varied greatly from the expectation. This was likely due to multicollinearity between the population and GDP per capita variables. GDP per capita is calculated by dividing city income by th eir metropolitan population explaining why including metropolitan population as an additional parameter was redundant. Figure 31 shows the relations ship between GDP per capita and metropolitan population. 52 PAGE 59 Figure 31 30,000 35,000 40,000 45,000 50,000 55,000 60,000 65,000 70,000 75,000 0 4,000,000 12,000,000 20,000,00 0 Met PopLBPCGDP (Scatterplot showing correlation between Pe r Capita GDP and Metropolitan Population) There appeared to be a directly proportional relationship between GDP per capita and metropolitan population, therefore both variab les were not included in the regression relating winning with revenue. The coefficient produced for DLWPCT wa s approximately 28.1 with a tstatistic of 4.35 generating a probability of 0.00 confirming statistical significance. The GDP per capita measure produced a very small coe fficient of 0.0008 whic h was statistically significant with a probab ility of 0.02. This would mean that a city's standard of living doesn't necessarily affect the amount of revenue that a team receives. The defensive lineman data produced a resu lt that is statistically significant and can be used in the next step of the analysis but to confirm this result another regression was performed using the linebacker data set. The results of Equation 28 were almost identical to the first. This was the goal of running a second regres sion relating winning to 53 PAGE 60 revenue. The m odel had an Fstatistic of 11.3 producing a probability value of 0.00. The R2 value of this regression was approximately 0.16 meaning that it differed from the first regression by 0.01. The coefficient produced by the linebackers' win percentage variable was 28.02 which only differed from before by 0.05. This coefficient produced a tstatistic of 4.32 making the probability value 0.00. The GDP per capita coefficient was once again 0.00 for the first several decimal points and had a probability of 0.03. This essentially confirms the legitimacy of the coefficient cr eated in the first regression; therefore, the coefficient used to related team revenue to player performance will be approximated to 28 for all positions to be determined. Since GDP per capita produced such a small coefficient in both instances, it won't be consid ered in the following parts of the analysis. Relating Gate Revenue to Winning The results of the Equation 29 were very similar to the results produced by the Equations 27 and 28. The Fstatistic was ap proximately 13.9 with a probability of 0.00. The R2 value for the regression was 0.18, mean ing that the fraction that winning contributes towards gate reve nue is identical to overall revenue. The coefficient for winning percentage was the only key difference in this analysis valued at approximately 15.6 with a probability of 0.00. The coefficient for GDP per capita was still very small and statistically significant; this value shoul d not have changed between the two different models. Thus, the conclusion of this regression is that gate revenue corresponds to just over half of team revenue. To confirm the results of the previous regression, the same model was calculated using the linebacker data set, which produced nearly identical results. There were no values in Equation 210 that differed from Equation 29 enough to be reported. Once coefficients were produc ed relating both play er performance to 54 PAGE 61 winning and relating winning towards both gate and overall revenue, the next step of the analysis could be conducted. This was dete rmining marginal revenue product for all of the athletes and performances produ ced across the various data sets. Monopsonistic Exploitation at the Quarterback Position Once marginal revenue product was determined for each quarterback's performance, each of these quarterbacks' 2008 salaries was recorded and compared to their marginal revenue products. The offensiv e data corresponded to a particular game rather than a season, so the calculated marg inal revenue products were divided by sixteen to show the amount the player deserved for a particular game. The values calculated were much higher than expected as the highest va lues had marginal revenue products of $7.4 million per game. Over the season this added up to $118.79 million for the season; however, this would be a circ umstance of a quarterback ha ving a perfect passer rating over the entire season. This hasn't come cl ose to being achieved si nce the highest rated passer over an entire seas on is Steve Young in 1984 with a passer rating of 112.8. The marginal revenue product of this performa nce in 2008 would have been approximately $84.6 million for the entire season. The mean marginal revenue product was $65.21 million and the minimum calculated marginal revenue product was $9.2 million. Figure 32 shows the relationship between quarterback MRP and salary. 55 PAGE 62 Figure 32 0 20 40 60 80 100 120 50 100 150 200 250 300 350 400 450 50 0 QB MRP QB SALARY Observations Dollars in MillionsQua r te r backMa r ginalRevenueP r oduct and Sala r y (Line graph comparing the results for quarterback MRP and actual salary) There are only a few observations towards the lower end of the data set where quarterback salary exceeds calculated MRP.8 The calculated rate of monopsonistic exploitation for quarterbacks was approxi mately 88%. The staggering amount of exploitation could be a result of quarterbacks having a much higher R2 term than the other regressions measuring performance. Running Backs Using the 0.18 coefficient produced in Equation 22, the MRP for running backs was calculated. Since Tatum Bell had the highest performance rating in the data set, he generated the greatest MRP valued at $54.7 million which would be $3.42 million for the game. Ryan Grant's performance against Tampa Bay was the worst calculated performance which was worth $264,000 that game. The average MRP for the running 8 All line graphs in this chapter are ordere d by calculated MRP in descending order. 56 PAGE 63 backs' performances was $15.1 million which would be about $940,000 each game. Instead of recording every r unning back's salary from 2008 to determine monopsonistic exploitation, representative salaries were recorded to aggregate the data. Figure 33 shows the relationship between the calculate MRP values and the represented salaries recorded. Figure 33 0 10 20 30 40 50 60 100 200 300 400 500 600 700 RBMRP RBSALRunningBackMa r ginalRevenueP r oductand Sala r y Observations Dollars in Millions (Line graph showing comparing the results for running back MRP and their represented salaries) The MRP values lie far above the represen ted salary values. The 1st, 2nd, and 3rd quartile values for running back MRP were $9.18M, $11.46M, and $19.28M, respectively. The MRP calculations were much higher than expected, therefore the rate of exploitation was much higher than expected. The rate of monopsonistic exploitation for running backs was evaluated at approximately 75%. 57 PAGE 64 Wide Receivers W ide receiver MRP was calculated using the same method as the running back position. Randy Moss's performance earned the highest marginal revenue product per game valued at $1.5 million with the mi nimum performance earning $64,000. Figure 34 shows the relationship between wide receiv er MRP and their represented salaries. Figure 34 0 5 10 15 20 25 100 200 300 400 500 600 700 800 900 1000 WR MRP WR SALARY Dollars in Millions Observations W ide Receive r Ma r ginalRevenueP r oduct and Sala r y (Line graph showing comparing the results fo r wide receiver MRP and their represented salaries) The overall median marginal revenue product was $5.65 million, the first quartile was $3.44 million, and the third quartile wa s valued at $9.33 million. Using these representations of MRP compared to salary, the rate of monopsonistic exploitation for wide receivers was valued at 48%. 58 PAGE 65 All of the offensive calculations appeared to be higher than e xpected which either means very high monopsonistic exploitation for offensive players, or the methods used to calculate the rate of exploitation affected the results. Defensive Linemen The defensive data sets consisted of data taken across the entire season, so the statistics put out were much more accurate in terms of representing marginal revenue product when compared to salary. Figure 35 shows the relationship between defensive linemen MRP and represented salaries. Figure 35 0 1 2 3 4 5 6 7 8 9 25 50 75 100 125 150 175 DL MRP DL SALARYDefensive Line m enMa r ginalRevenueP r oduct and Sala r y Dollars in Millions Observations (Line graph showing comparing the result s for defensive linemen MRP and their represented salaries) Salaries appear to exceed MRP at the higher end of the Figure 35, while the lower MRPs appear to be above actual sala ry. Dwight Freeney had the highest measured WPA for defensive linemen which translated into having the hi ghest calculated MRP 59 PAGE 66 m easured at $5.72 million. Freeney's teammate Josh Thomas had the lowest WPA which led to his marginal revenue product equaling $790 thousand. The mean MRP for defensive linemen was $2.50 million and the median was $2.26 million. The first quartile value was $1.63 million and the third quartile was $3.29 million. The rate of monopsonistic exploitation for defensive linemen evaluated at the represented areas was calculated to be approximately 10%. Linebackers Marginal revenue product and monopsonist ic exploitation were calculated for linebackers with the exact same method as defensive linemen. Figure 36 shows the linebacker relationship between MRP and salary. Figure 36 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 110 120 LB MRP LB SALARYLinebacke r Ma r ginalRevneueP r oductand Sala r y Dollars in Millions Observations (Line graph showing comparing the results for linebacker MRP and their represented salaries) 60 PAGE 67 The salaries seem to be above MRP at th e higher end of the data, but exploitation seems to occur at the lower end. This trend seems to be similar to the trend in the defensive linemen data. Jonathan Vilma had the highest measured WPA, thus he had the highest measured MRP worth $7.88 million. Th e lowest measured MRP for linebackers was D.D. Lewis' which was evaluated at approximately $445,000. James Harrison, the defensive player of the year, should have been paid $7.3 million, however he was paid much less since he was still under his rookie contract with the Steelers. The average linebacker should have been paid approxima tely $3.26 million. The rate of monopsonistic exploitation at the linebacker pos ition was evaluated at 8%. Secondary The Equation 26 was not statistically si gnificant, but MRP was still calculated for these players despite the results. Figure 37 shows the results of these calculations, they vary greatly from the other positions. 61 PAGE 68 Figure 37 0 2 4 6 8 10 12 14 25 50 75 100 125 150 175 Secondary MRP Secondary SALARYSeconda r y Ma r ginalRevenueP r oductand Sala r y Dollars in Millions Observations (Line graph showing comparing the result s for secondary MRP and their represented salaries) Salaries exceeded MRP at every section of the distribution other than the lowest. The results for MRP were far below the exp ectation with the most productive defensive back Will Allen only earning an MRP of $2. 4 million. The least productive secondary player measured earned an MRP of $119,000. The average MRP for this position was approximately $1 million. The first and third quartile values for MRP were $0.5 million and $1.21 million, respectively. The low figures for MRP resulted in a negative amount of monopsonistic exploitation equal to 446%. This was the only negative amount for average exploitation among the six positions, so it is likely that the lack of statistical significance played a role in this result. 62 PAGE 69 Results for Marginal Gate Revenue Gate Revenue was calculated to be appr oxim ately 54% of total revenue, meaning that the calculated MRPs for gate revenue were a linear extension of the calculated MRPs for total revenue. Tables 31 and 32 re veal this relationship by summarizing all calculations used to form conclusions from the analysis. Total monops onistic exploitation for standard revenue excluding the secondary results was measured at approximately 40% and total exploitation for gate revenue was approximately 12%. Table 31 Position Rating MRP GMRP Exploitation Gate Exploitation Max 158.3118.863.60.99 0.99 Third Quartile 105.879.442.50.93 0.87 Mean 86.965.234.90.88 0.76 Median 85.36434.30.92 0.84 First Quartile 69.752.327.90.89 0.81 Quarterbacks Min 12.39.24.91.44 3.56 Max 10.954.729.30.99 0.98 Third Quartile 3.819.310.30.95 0.9 Mean 315.18.10.75 0.53 Median 2.811.56.10.84 0.69 First Quartile 1.89.24.90.68 0.49 Running Backs Min 0.84.22.30.22 1.27 Max 14.724.717.60.98 0.97 Third Quartile 5.69.350.88 0.78 Mean 46.730.48 0.03 Median 3.45.71.90.76 0.55 First Quartile 23.41.80.65 0.34 Wide Receivers Min 0.610.92.03 4.67 Average of Means 29.0015.330.70 0.44 (Table summarizing all values measur ed in the offensive analysis) 63 PAGE 70 Table 32 Position WPA MRP GMRP Exploitation Gate Exploitation Max 1.815.73.20.91 0.84 Third Quartile 1.043.31.80.42 0.03 Mean 0.812.71.50.08 0.65 Median 0.722.31.30.13 0.55 First Quartile 0.521.60.90.18 1.13 Defensive Linemen Min 0.250.80.41.64 3.77 Max 2.37.94.30.81 0.65 Third Quartile 1.264.32.30.5 0.08 Mean 0.953.31.80.08 0.7 Median 0.933.21.70.32 0.25 First Quartile 0.5821.10.28 0.33 Linebackers Min 0.130.40.24.07 8.36 Max 22.71.40.39 0.07 Third Quartile 1.021.20.70.73 2 Mean 0.750.980.513.6 7 Median 0.670.880.461.8 3.9 First Quartile 0.420.50.35.1 9.6 Secondary Min 0.10.130.0718.5 33.1 Average of Means 2.331.271.15 2.78 Without Secondary 3.001.650.08 0.68 With Offensive Means 16.008.490.39 0.12 (Table summarizing values measured in the defensive analysis) 64 PAGE 71 Chapter 4: Conclusion Evaluating Results Five out of the six positions resu lted in significant statistical results showing a positive amount of monopsonistic exploitation. Averaging the rate of exploitation across the five positions indicates approximately 40 % total exploitation. This means that players are substantially underpaid in terms of their marginal re venue products. This was higher than the anticipated rate as the National F ootball League allows for collective bargaining with the players union. In labor market s, collective bargai ning and government intervention can both be used to cut the pow er of monopsonists (Boal and Ransom, 2002). During the 2011 NFL lockout, collective bargai ning failed as the players and owners could not make a compromise over the allocation of the $9 billion that the league earns in yearly revenues. After two extensions, th e previous collective bargaining agreement expired on March 12, 2011. This lead to th e player's union decertifying, allowing individual players to file class action lawsuits against the league. The lawsuits claim the lockout unlawful because it represents "a patently unlawful group boycott and pricefixing arrangement or, in the alternative, a unilaterallyimposed set of anticompetitive restrictions on player movement, free agency, and competitive market freedom." ( Tom Brady, et al. vs. NFL 2011). The players believe that th e owners are colluding to keep wages down, since wages are prices in a labor market. This collusion would violate the Sherman AntiTrust Act of 1890 since the 32 franchises act as one individual profit seeking unit. Historically, in America's major sports these types of law suits have generally favored the players primarily beginning with Flood v. Kuhn (407 U.S. 258) (1972) which 65 PAGE 72 helped es tablish free agency in professional baseball. Scully's article was published two years later, providing econometric proof of monopsonistic exploitation. This econometric proof played a role in the collective bargai ning that soon followed, since the reserve clause was abolishe d a year later. The results generated by this research represent statistically significant figures showing monopsonistic exploitation in the NFL. The concluding argument states that the players are correct in their ac cusations that the league is participating in pricefixing activities, as well as restricting free mobility of labor. The argument in baseball is that the teams invest substantially in their player s because they go through the minor league system after being drafted and, generally, are no t able to contribute substantially to their teams until many years after being drafted. Football does not have a minor league system and the majority of their training is perfor med at the collegiate level. There are key differences between playing football at the co llegiate and professional levels, but most of these are not teamspecific. Every club ha s their own playbook including offensive and defensive schemes, but the mental and physical differences between the two levels can be taught by any professional team. Thus, very limited investment occurs for each individual player and such investments were covere d in the previous collective bargaining agreement through restricted free agency. Money is the predominant reason for labor issues that have occurred in professional sports and the current situa tion is no exception. Originally, NFL owners opted out of the previous CBA because they felt it favored the players. One aspect of the negotiations that both players and owners have agreed on pertains to rookie contracts, particularly for first round draft picks. Fi rst round draft picks ge t paid considerable 66 PAGE 73 am ounts in salary without having to prove th emselves on the field first. The 2008 draft's first overall pick, Jake Long, was paid approx imately $6 million in salary. The structure of NFL contracts shows his salary will increa se as his contract matures (USA TODAY, 2011). The owners also wanted to establish an 18 game regular season to increase revenue mostly by increases in gate and br oadcasting revenues. The players oppose this proposition, claiming that the current length of the season takes a physical toll on them, both short term and long term. Hence, medical and retirement benefits for veterans are controversial when negotiating a new collective barg aining agreement. These sectors of the negotiations were minor compared to th e biggest problem of allocating the league revenue. The previous CBA was established prior to the 2006 league year and was designed to expire prior to the 2012 league year. However, the owners exercised their option to opt out of the last year of th e CBA before the 2008 season (Clayton, 2008). The data measured in this analysis pertained to the 2008 season, hence it occurred after the owners opted out of the past CBA. However, monopsonistic exploitation was still detected and proven using econometric methods. This implies that th e owners should be satisfied with the agreement reached in 2006 and leaves them vulnerable to receiving a smaller portion of league revenues as a resu lt of further negotiation or a ruling in the Tom Brady et al. v. National Football League, et al. case (2011). This work has shown that the owners were able to exercise monopsonistic exploitation that held down the salaries of NFL players in the 2008 league year at a rate of approximately 40%. The results from the gate revenue calculations lead to different conclusions. The amount of monopsonistic exploita tion from gate revenue was calculated to be 12%. This 67 PAGE 74 value im plies that if gate revenue is the onl y revenue to which players contribute, than players have more bargaining power than owne rs. Portions of gate and local revenue are the only streams of income not shared among the 32 owners in the league, implying that this could be the only possible revenue for which winning could contribute. The rest of the revenue is shared in the league, thus ow ners would receive it regardless of their team's winning percentage. This conflict represents another issue of performing this type of analysis for professional football, because the NFL uses more revenue sharing than the other major American sports leagues (Fort, 2004). Winning does not directly affect the majority of team revenue as calculated in this process. However, contributing towards wins is the primary method that determines an athlete's salary. Professional football is a labor intensive i ndustry, because the players are the entertainers and sellers of the product. The method used to establish players' MRP in this work may not apply directly, due to NFL revenue sharing. However, since winning is their primary salary determinant and footba ll is the product to which they contribute towards, player's contributions towards sh ared league revenue and individual team revenues are applicable to calculating their MRP. Thus, the more accurate determination of MRP and monopsonistic exploitation calculated in this work would be total team revenue and not just gate revenue. Improvements to the Analysis The process used to calcu late exploitation was successful in producing statistically significant results, but there ar e many areas where different methods could have improved accuracy of the results. Exam ining one year's results may not have completely represented the microeconomic structure of the labor market in NFL, 68 PAGE 75 especially when the majority of this season was played in the fall of 2008 during the recen t financial crisis. The crisis would have caused a negative impact on revenues for this season in the form of fewer ticket sale s and other streams of revenue. This could mean that the marginal revenue products for football players are us ually higher than they were calculated in 2008, since re venues that year were lik ely decreased, although average franchise value was shown to increase slightl y, but much less when compared to the other major American sports leagues (Humphreys, 2010). One factor that seemed to skew the offensive calculations was that each observation pertained to one performance in a single game while the defensive data represented performance across the season. When calculating for the entire season, small issues such as injuries are taken into account. Also, in many cases, better and worse offensive performances, particularly quarter backs, would not be replicated over the course of a season leading to a normally distributed data set. Quarterbacks Many discrepancies in the calculations occurred for each position, including quarterbacks. Many of their contributions, both positive and negative, were likely left out since only QB rating was used to evaluate th eir performance. The QB rating is a very accurate statistic to measure a quarterback's performance throwing the football; however, it leaves out a growing facet of the quarterback game, ef fectively running the football. Tyler Thigpen of the Kansas City Chiefs rushed for 386 yards and three touchdowns in 2008 and David Garrard, who plays for the Jacksonville Jaguars, ran for 322 yards and two touchdowns over the course of the s eason (NFL Enterprises LLC, 2009). Many running backs are not able to accrue this many yards over the course of a season, 69 PAGE 76 especially if they have to split rushing opportuniti es with another running back. Thigpen and Garrard are able to accrue this many rushing yards, despite playing the quarterback position. These types of hybrid players are difficult to measure in terms of their contributions because one variable usually does not measure their performance enough, while multiple variables could cause problem s with collinearity in the regressions. Negative factors exist that the QB rating om its, including fumbles and sacks. However, interceptions are negative fact ors included in QB rating; a qu arterback losing a fumble to the other team is just as detrimental to the outcome of a game as an interception. Another small factor that could improve the evaluati on of the passer rating would be to discount throws that were accurate but the wide receiver dropped the pass. However, something like this would leave too much r oom for gray areas in an attempt to establish fault of the play between passer and receiver. Quarterbacks were shown to have an esti mated rate of monops onistic exploitation of approximately 88%. Meani ng that, quarterbacks should be paid 8.3 times more than they were in 2008.9 Thus, the work suggests that play ers at the quarterback position are substantially underpaid. Running Backs The goal of the running back rating crea ted was to accurately measure how well a runner's performance contributes toward winni ng when compared to the rest of the observations. The method that th e statistics were weighted seemed to stress emphasis on certain aspects of a runners ability. The ne t scores were divided by weighted number of attempts to create the gross score, the data seemed to give heavy emphasis on yards per 9 The proper salary approximation figures were derived by an extension of Equation 212 using general examples presuming the calculated rate of exploitation. 70 PAGE 77 rush. This statistic can be misleading because better running backs usually receive more hand offs, lowering their yards per rush, part icularly near the end of a game when a winning team is running the football to run off the clock. These rushes typically do not go for many yards. Some of the key aspects left out of this rating were a running back's contributions in pass blocking, pass receptions, special team contributions, and the impact of the offensive line. In pass blocking, particularly in shotgun formations, a running back will remain in the backfield during a passing play where he serves as th e last blocker before the defenders reach the quarterback. This role includes throwing their body at an interior defender that made it past the offensive line or picking up a defender blitzing from the side of the line. Blocking c ontributions would be difficult to quantify since the statistics of a block are not recorded e xplicitly. One could only measure the result of plays near the blocker. As a result, an adjustment on a runner's perf ormance based on blocking was not included in the RB rating. Subsequently, this is why offensive linemen were not included as a position measured in this work The inclusion of receiving yards as a running back's contributions would not have been as difficult to quantify as the blocking aspect, but it was not a necessity because receiving is not the primary role of a running back. As a result, a receiving running back's abilities may be undervalued due to their effectiveness in the passing game. Running backs were calculated to be expl oited at a rate of 75%, meaning that running backs should be paid on average four ti mes more than their cu rrent salaries. This amount shows the NFL running backs are vastly underpaid for their onfield contributions. 71 PAGE 78 Wide Receivers Wide receiver performance was also measured using a score generated in this analysis based on different receiver cont ributions. The receiver score seemed more accurate than the running back score. Some of the NFLs most renowned wide receivers stood on top of the data set including Randy Moss, Brandon Marshall, and Anquan Boldin. During the developmental stages of determining a method to evaluate a wide receiver, an issue kept arisi ng where the best performers were speed receivers who would have one reception for many yards and a t ouchdown. While plays lik e these radically affect a game's outcome, they do not show how valuable the wide receiver is overall. The reception statistic was altered to become an output factor rather than a divisor of performance increasing the accuracy of the wide receiver rating. Th e rationality behind this change was that when a receiver catche s a pass, he has succeeded in breaking away from coverage during his route allowing the qu arterback to feel confident throwing him the football. The best receivers are those that a quarterback feels more comfortable throwing to often, which was reflected in th e calculation of wide receiver rating. The WR rating was still not all encompa ssing because of significant special teams contributions made by fast wide receivers that were unaccounted for. Faster receivers are often asked to return kick offs due to thei r ability to make defenders miss tackles and rapidly accelerate after stopping or switching dir ections. Another factor the wide receiver score left out was number of dropped pass es. Dropped passes play a large role in determining how valuable a receiver is to a team. The limitations arise because dropped 72 PAGE 79 passes are not one of the readily recorded st atistics in an NFL game. Additionally, wide receivers also provide minor blocking contributions. One common example is when a run goes outside of the offensive line towards th e edge. A receiver is then required to block the cornerback playing on that side so the r unning back is able to maneuver forward. One more example would be when one receiver cat ches a pass near another and the receiver without the ball will attempt to block the de fensive backs downfield for their teammate. The more physical receivers are best at accomplishing these types of blocks. Out of the three offensive positions measur ed, the wide receiver results appear to be the most accurate regarding measurement of marginal revenue product and monopsonistic exploitation. This could be due to the score that was created or the large number of observations in the data set. Ther e is a direct correlat ion between amount of statistical significance and number of observatio ns or degrees of freedom. Wide receivers were determined to be exploited at a rate of 48%, which is closest to the average amount of exploitation out of the six positions. This rate implies wide receivers should be paid 1.9 times more than they are currently paid. Defensive Linemen and Linebackers The WPA variable used to measure defens ive performance was chosen because it directly shows a player's effect on winni ng. Defensive player contributions are very difficult to quantify as these uni ts are highly collaborative, so these evaluations can be either over or underestimated. Some statistics are positive in every instance such as sacks, fumbles recovered, and interceptions, but many other statistics, such as tackles or passes defended, can be misleading. Like any statistic that attempts to generate an all encompassing evaluation of the player perf ormance, WPA has its weaknesses. When a 73 PAGE 80 player m akes a touchdown saving tackle, then they contributed on a pl ay that lowered the probability of winning even though they saved the probability of winning from decreasing further. A better defensive player evaluation w ould include the number of downs that a player was on the field. This would help to create a data envelopment analysis using downs played as the input and the standard defensive measures as the outputs (DeOliveira and Callum, 2004) Additionally, this would help to compare the contributions of second or third string player s to the starters. Overall, WPA seemed to accurately evaluate the performance of defe nsive linemen and linebackers to calculate their MRP and rate of exploitation. Defensive linemen and linebackers were cal culated to be exploited at a rate of approximately 10% and 8%, respectively. This means that players at these positions deserve 1.1 times more paymen t than currently received. Secondary The issues quantifying contributions of players in the defensive backfield have been noted. Cornerbacks and safeties top the list of positions in the category of misleading statistics. A cornerback playing hi s position perfectly does not show up in the statistics column unless he is stopping a run or tackling a wide receiver behind the line on a screen pass. The results for the secondary pos itions were most likely not significant for this reason. To calculate marginal revenue product and achieve statistically significant results, a cornerback's ability to prevent their man fr om being thrown the ball would need to be included. A suggestion would be measuring the success rate of cornerbacks. Success rate 74 PAGE 81 75 "is the percentage of passes that don't manage to get at least 45 pe rcent of needed yards on first down, 60 percent of needed yards on second down, or 100 percent of needed yards on third down." (Schatz, 2011). Anothe r method used to ra nk cornerbacks is average yards per pass on charted targets, meaning the average yards gained on plays where a quarterback targeted a receiver that was matched up with the cornerback being measured. These different statistics woul d likely produce better evaluations of a cornerback's contribution. Using a safety's standard statistics to measure their performance would not be nearly as misleading as with cornerbacks. Safeties often come over to help out a cornerback who has lost c overage on his man, making their contribution positive. A safety in man coverage applies double cove rage on a troubling re ceiver or guarding a running back. The safetys role is to make a play when it needs to be made; thus, evaluating them based on standard statistics would work. WPA is probably succinct for measuring safeties, especially considering that WPA is able to account for high pressure situations, when safeties are commonly involved. A better method for evaluating performance at the secondary position would be to divide co rnerbacks and safeties into two different data sets. For statistical signif icance to be obtained, more than one season's data would be necessary. The performance va riable for cornerbacks would either be success rate or average yards per pass on target s; one could include both if the variables did not correlate too high with one another. For safeties, a regression including tackles, interceptions, defended passes, and forced fumbles could work or WPA. The method including the four parameters just described could be perf ormed at the individual game level so that only one season's data would need to be recorded. These improvements PAGE 82 would likely produce statistically significant results to accurately evaluate m arginal revenue product at the cornerback and safety positions. Discussion The calculations produced by this work su ccessfully showed a positive amount of monopsonistic exploitation for the majority of the players measured. While the methods mostly featured a team's starters, additional research benefit evaluating the contributions of players who do not play as often. Possible regressors in such an analysis could be when a player was picked in a draft, experi ence in the league, or a possible marketability factor. Moreover, further research in determining rookie salaries would prove immediately beneficial in the NFL as it remains a highly debated subject in collective bargaining. Additionally, quantifying player ma rketability would prove beneficial as it can be an additional contribution to team revenue. An additional accomplishment in this work was the generation of a value to determine the overall performance of running backs and wide receivers, particularly the wide receiver score generated could prove bene ficial to those attempting to rank receivers as it seemed to accurately m easure their performances. As the NFL continues with their labor st ruggles, the results provided in this work would serve as an asset on the player's side of negotiating and co uld help to end the current lockout. 76 PAGE 83 References Boal, W.M., & Ransom, M.R. (2002). Monopsony in American Labor Markets. In R. Whaples (Ed.), EH.Net Encyclopedia Retrieved from eh.net/encyclopedia/article/boal.monopsony Bureau of Economic Analysis (2011). Regi onal Economic Accounts Charting [Data File] Retrieved from: http://www.bea.gov/regional/index.htm#gsp Burke, B. (2010, January 27). Win Probability Added (WPA) Explained [Web log post]. 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