The core economic process within any professional sports team is the sporting production process in which ‘raw’ playing talent is transformed into player and team performance and, in turn, match outcomes. Formally, this process can be modelled as a sporting production function. (Gerrard 2001, European Sport Management Quarterly, 1: 3 p220)
One of the first empirical applications of production functions in sport was presented by Scully (1974) on monopoly exploitation in baseball. Scully (1974) aimed “to crudely measure the economic loss to the players due to the restrictions of the reserve clause” (Scully, 1974, p915) and set out to accomplish this by obtaining estimates of the marginal revenue product of pitchers and hitters performing in the MLB, before
comparing these estimates to player earnings in order to determine what percentage of a player’s MRP was retained by him.
Scully (1974) proposed a causal chain (recursive) model consisting of two equations, the first of which directly determined the marginal effect of player performances upon team success and the second the marginal effect of team success upon team revenues. Scully (1974) adopted an OLS regression estimation method and applied it to both equations in order to estimate these marginal effects, which in turn enabled Scully (1974) to estimate player MRP.
The first equation of Scully’s (1974) model consisted of the dependent variable PCTWIN (a calculation of the percentage of team wins), which Scully hypothesised to be a measure of team success, and several other independent variables. The independent variables included measures of player performance such as the team slugging average (TSA) and team strikeout-to-walk ratio (TSW). TSA was hypothesised to be a measure of batting (attacking) prowess and measured the number of bases gained per match whereas TSW measures a pitchers ability to control pitches and was therefore used a measure of the pitching (defensive) contribution to team wins.
In the second stage of Scully’s (1974) model, PCTWIN (now specified as an explanatory variable) and other independent variables that accounted for local population, team fan base, stadium location and racial prejudice were used to determine team revenues. Scully (1974) then used the coefficients obtained for TSA and TSW (from the first stage) in conjunction with the coefficient obtained for PCTWIN (from the second stage) to estimate average MRP. In order to convert these average MRP estimates to MRP per player Scully (1974) assumed that the team’s performance was the sum of the individual’s performances. Furthermore, Scully’s estimations did not incorporate a team’s full roster but only twelve out of fifteen hitters (that were expected to have played in a match) and eight out of ten pitchers (expected to have played in a match). Scully (1974) then presented ‘gross’ MRPs that did not account for costs associated with placing players on the field (i.e. cost of training players, cost of equipment and transport costs) before calculating ‘net’ MRPs after the deduction of these costs.
Finally, Scully (1974) estimated player salaries that would have been paid to players (in various categories of skill) in order to determine whether players were remunerated in accordance with their MRP or if they were exploited. These estimates were obtained from separate salary regressions for 148 players (hitters and pitchers) performing in the 1968 and 1969 seasons. Scully’s (1974) results ultimately suggested baseball players were being exploited to a fairly high degree.
Scully’s player performance model has been criticised by others. Medoff (1976) for instance argues Scully’s (1974) model was incorrectly specified as the recursive model did not account for simultaneity between win percentage and revenues in the two equations. Consequently, Medoff (1976) conducted a follow up study utilising a Two Stage Least Squares estimator and revealed similar results that confirmed the exploitation of baseball players. Furthermore, Gerrard (2001) highlights the limitations of Scully’s model by claiming the model is only applicable in those team sports where comprehensive player performance data can be captured easily, such as batting and fielding games like baseball and cricket.
In another study on baseball, Zech (1981) employed a Cobb- Douglas production function16 in order to identify those factors that contributed to team success in the MLB. Once identified, Zech (1981) empirically estimated the effect of these factors and used the results of the empirical testing to construct a measure of the league’s most valuable player (MVP).
Zech (1981) summarised player skills into four main categories, hitting, running, defence and pitching. Measures used to account for these player skills included, players batting averages (as measures of hitting frequency), number of homeruns (as measures of player power), a teams stolen base total (as a measure of team speed), a players fielding percentage (as a measure of catches taken), total chances taken (as a measure of difficult catches taken) and pitchers strikeout to walk ratio (as a measure of pitcher ability). In addition, Zech (1981) also incorporated two measures of head coach ability in the form of a manager’s lifetime won-lost percentage and the number of years managed in the major leagues.
16 A Cobb-Douglas function is a common function used by economists when turning to empirical work (Gartner 2006). The
Zech (1981) established a baseball production function and then uses it to compute each player’s marginal product by calculating each player’s contribution to team victories. The top ten most valuable players’ (MVPs) in the league computed by Zech (1981) differed significantly from the official voting list of sportswriters (who selected MVP’s for each season using an informal weighting system for each of the player’s different skills). Zech (1981) suggests this discrepancy may have been due to player publicity and personality playing a role in the sportswriters’ weighting schemes.
Zech (1981) tested his model using data from 26 MLB teams competing in the 1977 baseball season and tested the model four times employing all possible combinations of defensive and managerial ability in order to ensure no multi-collinearity. Ultimately, Zech (1981) revealed that no measures of a team’s defensive skills (player fielding percentage and total chances taken) were significant and the contribution of the manager was also revealed to be insignificant. Furthermore, hitting was revealed to contribute almost 6 times as much to team success than pitching and the number of homeruns contributed about as twice as much as the number of stolen bases to a team’s success.
Similar methods of modelling the sporting production process implemented by Scully (1974) and Zech (1981) have since been applied to other sports. Schofield (1988) estimated production functions in cricket using data from the 1981 to 1983 cricket seasons in the County Championships and John Player League in England. Similarly to the studies detailed above Schofield (1988) specified output to be team success and a function of player performance and weather. Player performance (measured by batting, bowling, captaincy and fielding variables) was further composed of a sub-set of variables namely: coaching skills, team management skills, player availability, scouting, form, experience, the quality of practice and training facilities. Schofield (1988) therefore adopted a recursive system in which endogenous variables of player performances and team success were sequentially determined.
Schofield’s (1988) study involved collecting cross sectional data from seventeen competing teams and data was initially normalised to account for random variation in the mean caused by external factors that had an influential impact on input and output
first measure was the number of wins achieved throughout a single cricket season and the second measure was the number of points acquired. Schofield’s (1988) analysis was broken into three parts; the first part assessed the general importance of batting and bowling and the second and third parts assessed the relative importance of different types of batting and bowling skills.
Batting performance was measured by calculating the number of runs scored per game and per over and the bowling variables included in the production function included the calculation of a bowling average (runs conceded/wickets taken) a composite measure of both attacking and defensive bowling skills. Furthermore separate measures of attacking bowling such as the number of wickets taken per game and number of overs bowled per wicket taken as well as measures of defensive bowling such as the number of runs conceded per over were also included as bowling inputs. Using these calculated values Schofield (1988) developed a linear production function using OLS and ultimately concluded that generally bowling had a greater impact upon match success than batting.
Bairam et al (1990) also constructed a cricket production function in their study on cricket played in Australia and New Zealand. Although, this study was largely based on Schofield’s (1988) study it differed in a few key areas. Output was defined as points percentage rather than number of wins or points achieved. Furthermore, two measures of bowling ability were utilised in this study, the first a measure of attacking bowling, balls per wicket and the second measure of defensive bowling, the number of runs scored by the opposition. Runs scored per wicket was another dimension that was included and reflected attacking and defensive bowling. Runs per innings and runs per over were two calculated values that represented the batting performance of teams.
Bairam et al (1990) chose to omit the element of fielding in order to avoid multi- collinearity. Pooled data was collected and normalised by taking ratios of the variables used to their seasonal means and multiplying the ratios by 100. This was done to account for random events such as differences in weather conditions that could have influenced both input and output variables. Bairam et al (1990) revealed that both batting and bowling variables were important in explaining match success.
In order to maximise the probability of match success, Bairam et al (1990) concluded that New Zealand should adopt a combination of attacking batting and attacking bowling strategies. In a comparative analysis with England it was revealed that relative to England, batting was marginally more important than bowling for New Zealand. The results obtained for Australian cricket suggested the probability of match victory was maximised by adopting an attacking batting strategy and a defensive bowling strategy (Bairam et al 1990).
Carmichael and Thomas (1995) formulated a production function for rugby league football in the form of a recursive model in which both player performances and team success were determined sequentially. Team success (the output) was stated as a function of three measures of attacking performance, tries for, kicked goals for (conversions following a try or penalty) and drop goals for and three measures of defensive performance, tries against, kicked goals against and drop goals against. Carmichael and Thomas (1995) assumed these six performance inputs were influenced by variables such as player fitness, inherent ability, player experience, team organisation and coaching skill.
Output was measured by the percentage of games won during the season (expressed relative to the divisional average). Attacking performance was measured by the average number of tries, kicked goals and drop goals scored per game and defensive performance was similarly measured by the average number of tries, kicked goals and drop goals scored against. Carmichael and Thomas (1995) included height and weight variables as a proxy for player strength and measured experience by squaring the average player age. Inherent ability was proxied by the percentage of full international players and the percentage of players from overseas. The average number of appearances made per squad member measured team organisation and the ability of the coach was proxied by the number of years spent as a full time rugby coach prior to the season of investigation.
Carmichael and Thomas (1995) tested their model on cross sectional data collected for 35 rugby league clubs competing in the 1990-1991 season. Data was collected for 1,214 individual players and team input and output measures were also calculated as seasonal totals. In conclusion Carmichael and Thomas (1995) stated successful
attacking performance and suggested a disaggregated data set with more player specific data could potentially produce improved analysis of production functions in rugby league.
Carmichael et al (2000, 2001, 2011) and Carmichael and Thomas (2005) have conducted production function studies on English Premier League football (EPL). Carmichael et al (2000) employed disaggregated player performance statistics for matches played by teams during a single season (1997-98) in order to estimate a production function for match performances. The output was specified as the match score and was expressed as a goal differential as this measurement allowed for the easy interpretation of ordinary least squares (OLS) estimates (Carmichael et al 2000). The independent (input) variables consisted of various attacking and defensive play actions and team characteristics. Measures of play included; shots on target, % of successful passes, the number of tackles made, clearances, blocks, interceptions, dribbles won/lost, the number of free kicks conceded, yellow and red cards. Measures of the goalkeepers’ performance such as % of successful distributions and number of catches/drops were also included as independent variables. Ultimately, the results obtained by Carmichael et al (2000) emphasised the influence of attacking skills such as accurate shooting and passing as well as the importance of defensive skills as reflected in tackles, clearances and blocks made upon match outcomes.
In a follow up study Carmichael et al (2001) utilised aggregated match plays for each team over a full league season (1997-98) in order to examine team performance and efficiency by estimating a season based production function. Carmichael et al (2001) adopted a multiple equation (recursive) system where output (overall team success) expressed as percentage points was determined sequentially. The results obtained from this analysis emphasised the importance of defensive aspects of play and identified shots on the opposition goal, goals scored and accurate passing to be of particular significance with regards to output determination.
Carmichael and Thomas (2005) use match play statistics from the 1997-98 EPL season in order to examine the home field effects (advantages) on within match performance of home and away teams. A variety of team play measures are included as inputs in a team production function in which the number of shots and goals scored by each team in each match are specified as outputs. The model utilised by
Carmichael and Thomas (2005) incorporates a recursive system, with a team’s success in terms of match plays determined sequentially. Team play measures included play actions such as dribbles, runs, passes, fouls, tackles, clearances, blocks and saves made by the goalkeeper. The regression results revealed the statistical significance of attack related team play measures was stronger for the home team and the significance of defence related measures were stronger for the away team, implying attacking play is more important at home and defensive play more important away.
Carmichael et al (2011) study team performance in the EPL over the seasons 1998-99 to 2004-05 and investigate the link between playing success and commercial success. Employing a data set that combines financial measures and performance data, Carmichael et al (2011) based their empirical analysis on three behavioural equations. The first equation is a sports production function where league success (output) was interpreted as a function of playing inputs (performance statistics) and non-playing inputs such as managerial contribution and home support. The second equation was a standard revenue equation in which revenue was specified as a function of output and managerial specific inputs. Equation three was a hedonic17 wage-price or earnings function that related player performances to wage expenditure. Carmichael et al (2011) revealed on-field success to be directly related to player skills and abilities and revenue was positively influenced by on field success.
In a study on the EURO 2004 international football tournament, Carmichael and Thomas (2008) test an aggregated production function model for competing teams using a match play data set. Carmichael and Thomas (2008) constructed four play variables designed to capture the quantity and quality of attacking and defensive plays and included these variables as independent variables in a regression where the dependent variable was a team’s average goal difference throughout the tournament. In addition to the estimated average production function, Carmichael and Thomas (2008) also estimated a production possibility frontier in order to examine team efficiency. The use of production frontiers in order to measure team efficiency is also evident in literature on team performance. The sub section 3.4.2 below describes the
production frontier approach and presents the findings of some studies that have adopted this approach.