Abstract
Using panels of player pay and performance from Major League Baseball (MLB), we examine trends in player productivity and salaries as players age. Pooling players of all ability levels leads to a systematic bias in regression coefficients. After addressing this problem by dividing players into talent quintiles, we find that the best players peak about 2 years later than marginal players, and development and depreciation of performance appear to be more pronounced for players with the highest ability levels. Within-career variation, however, is less pronounced than between-player variation, and the performance level of players within a given quintile will typically remain lower than the talent level for rookies in the next higher quintile. We also find preliminary evidence that free agents are paid proportionately to their production at all ability levels, whereas young players’ salaries are suppressed by similar amounts.
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Notes
In baseball, the majority of this research stems from the seminal work of Scully (1974). Kahn (1991) summarized the early racial discrimination literature, which has since expanded to include analyses of coaching discrimination in MLB (Singell 1991), the NFL (Madden 2004), and the NBA (Humphries 2000; Kahn 2006). Marburger (1994, 2004) looks at final offer arbitration in MLB, while Zimbalist (1992), Kahn (1993), Vrooman (1996) and Miller (2000), among many others, look at the salary effects of free agency in baseball.
We note this technique is not the same as quantile regression, although both terms have a common element of placing greater analytical weight on particular centiles of a distribution. While it would be natural to consider using quantile regression, utilizing this technique would result in an observational bias in the same manner as would occur from using OLS. At very young (and very old ages), only the highest ability players will be in the sample, and thus, say the 50th percentile of the conditional distribution of performance at, say age 19 will be estimated using information from only these high ability players (whose median would likely be a q5 player). However, the 50th percentile at, say age 28, will be estimated utilizing information from players of different abilities (whose median would likely be a q3 player). The result is a dampening of the distinctions between groups of players with different abilities, which is precisely what we attempt to avoid by dividing players into quintiles. For an early example that uses quantile regression to examine salary discrimination in the NBA, see Hamilton (1997). More recently, Berri and Simmons (2009) and Vincent and Eastman (2009) utilize quantile regression to examine discrimination in the NFL and NHL, respectively. To the extent that entry barriers in college football make entry age into the NFL more homogeneous, the compositional effect will be reduced.
Many players in baseball make frequent journeys back and forth between the top minor league level (AAA) and “The Show,” earning them the label of “4-A” players.
As we note above, segmenting the sample reduces but does not eliminate the bias associated with players of different ability levels. Further increasing the number of groups would reduce the bias further, but would reduce the sample size within each cohort and the number of observations of performance for each age level. We choose five groups somewhat arbitrarily bearing sample size considerations in mind.
Our omission of the traditional set of controls from the Scully (1974) style models is intentional. We seek to look past the structural sources of pay variation and instead simply describe the pattern of pay variation for player groups similar to those in Eq. 1 and not, at this stage, to identify causative factors. Any resulting “omitted variable bias” is similarly intentional, as our objective is to compare and contrast the dynamics for performance and for pay, to confirm the general similarity, and to note whether there are systematic and large magnitude differences at particular ages.
The remaining distinctions we have added are to control for different player “markets”. The division of players into ability quintiles allows a crude control for ability, as a star quality player and a journeyman are not close substitutes. We include indicator variables for position both because there are separate markets for shortstops and first basemen, for example, between which there is limited ability to substitute, and also as a concession to identifying player MPP. Players with similar offensive productivity at more challenging defensive positions are demonstrating ability along another dimension and should merit a salary premium.
As could be predicted, the effect of the switch from age to experience in productivity Eq. 1 is that higher-ability players peak later in their careers, as they debut at younger ages, allowing more playing time prior to their peak age.
As noted above, we would prefer to estimate career paths of performance relative to age, and ultimately wish to compare the evolution of pay and performance across a common metric. We illustrate below that either age or experience can explain essentially the same fraction of variation in performance paths; a switch from using age to experience reduces R 2 by an average of 0.0002 across the five models (where each quintile is modeled separately). However, when we later attempt to predict salary paths, a switch from using age to experience increases R 2 by an average of over 0.15 across the five models. In using experience, we accept the reduction in ease of interpretation in performance in exchange for the enhanced efficiency of the salary estimations.
Fort (1992) estimated separate parabolic arcs for salary trends of players of below mean and above mean age. Although our differing arcs “hinge” at the point of free agent eligibility rather than mean age, the econometric objective is the same.
This may be because star players refuse to accept salary cuts as their contracts come up for renewal after a performance decline, leading to observational selection with a short lag, or that perhaps the marginal revenue product (MRP) of star players—the theoretical source of employer willingness-to-pay for the player’s services—does not decline as ability does. The latter possibility allows for veteran fan favorites and stars that either retain their popularity and drawing power through the end of their careers, or watch their fame decline more slowly than their fading skills, as was the case with the end-of-career versions of Cal Ripken, Jr. and Willie Mays. Although the discussion is beyond the scope of this paper, there is also a large literature in personnel economics that studies reasons why salaries may vary from MRP under competition. Common theories of this sort include efficiency wages to combat shirking and incentives for career contracting.
The Baseball Archive database is available at http://www.baseball1.com.
Defensive ability has proven both difficult to measure and, consequently, difficult to establish as a significant predictor of salaries except (weakly) through simple binary variables for defensive position.
James (1988) concludes there is very little evidence of team complementarities in terms of batter “protection” and line-up effects. While batting order has a marginal effect on count statistics such as RBIs and runs scored (and this is why these statistics are poor choices to measure hitting performance), teammates and lineups do not statistically affect a player’s underlying hitting productivity, which is what we use as a performance measure (OPS).
Indexed OPS controls for systematic changes in the league-wide OPS level at each position across seasons. The use of the third-best season is somewhat arbitrary, but serves three purposes. First, it avoids the overestimation of ability of players who did extremely well in an injury-shortened season or a partial-season “cup of coffee” call up from the minors. Second, the cutoff removes very marginal players with fewer than three qualifying seasons from further consideration in the sample. Third, it reduces the likelihood that a player with a positive outlier season is assigned to the wrong ability quintile. To further limit the sample, players younger than age 29 in 2005 are removed due to the possibility that they might not have peaked yet, as their inclusion might taint the cohorts. Our results our not sensitive to this assignment mechanism, and using the 2nd or the 4th season to identify peak ability yields very similar results.
The seemingly arbitrary threshold of 130 at bats is chosen based on the rules establishing “rookie status” by MLB (for the purpose of awarding the Rookie of the Year award). A player who has logged fewer than 130 at bats is still considered a rookie in the following season.
The cutoff levels of indexed OPS are 0.958, 1.023, 1.073, and 1.153. The indexed values are relative to averages which condition for year and defensive position. The use of player-seasons as the unit of measure, coupled with the longer career lengths of high-ability players, means that there are fewer players in the top (5th) quintile than in lower ones. There are 126 players in 5th quintile and 285 in the 1st quintile. There are slightly more than 1200 player-seasons in each quintile.
So as not to overstate this finding, it should be noted that the estimated raw OPS levels for top players at ages 26 through 33 are all within 15 points of one another, and given the stochastic error in year-to-year performance these fits could quite easily be thought of as statistically equivalent (the peak resembling a plateau more than a point). Also, the Chow test in Model 3 fails to reject the null that players in different quintiles have statistically equivalent aging patterns aside from having differential quintile intercepts.
The apparent similarity in age of MLB debut across quintiles may be slightly exaggerated in Fig. 2. While players in lower quintiles might be brought up as part-time players while young, and play their way into a regular job over two or three seasons, this is less common for top prospects who are projected to become stars. Because free agent eligibility is determined by time on a major league roster rather than playing time, having a young player sit on the major league bench has an opportunity cost of future productivity rents. This opportunity cost will be higher for players who are far above replacement level, so teams are reluctant to promote them until there is a starting position open for them. So while all quintiles are being called up at similar ages, the early observations for the top quintiles tend to represent much more playing time, and have more weight in the Table 2 regressions.
The mean and standard deviation of age at debut for the highest ability players (q5) are 21.6 and 1.6 years of age, respectively, while the corresponding measures for the lowest ability players (q1) are 23.4 and 1.9 years.
Implications (c) and (d) could both be formally tested using forecasting techniques that project the estimated regression equation outside the data range. All of those techniques, in one way or another, factor the extent of data extrapolation into their forecasting errors and confidence intervals, and in this manner would be conducive to formal hypothesis testing even at age and experience levels where the are no observations. We defer from using this approach, however, as the nature of our project is inherently empirical and we wish to avoid the assumption, necessary for extrapolation or forecasting, that the structural form of productivity growth or decline is unchanged outside our data range.
As an anonymous referee has pointed out, the 3rd best season might be difficult to operationally utilize to forecast the ability level of a young player. At the very least, such an algorithm would require three qualifying (>130 AB) seasons for the player. While the focus of our paper is not to predict the ability level of a player based on playing statistics, we still contend that such players could be identified at an early age. For example, a slight modification of our methodology, using the 2nd best qualifying season up to and including the season where the player’s (opening day) age was 25 would allow for fairly accurate prediction of a player’s eventual talent quintile. While this prediction would be limited to players who have had two qualifying seasons by age 25, who are disproportionately high ability players, of those player assigned to q5 by the alternative metric, 78.6% are ultimately assigned to q5 by the method described in the paper (3rd best of all seasons).
In the MLB collective bargaining agreement, experience is measured in days of service on a team’s roster, but data on service time is not publicly available. We estimate experience crudely by subtracting the player’s debut year from the season year. We set the cutoff at 7 years rather than 6 both due to the frequent practice of short end-of-season callups for young players, which will lead to an early debut but little service time, and as this cutoff maximizes goodness of fit in our later regression models.
The overall upward trend in salaries is not monotonic, interrupted briefly by collusion, expansion, and other short-term influences, and is faster than the rate of inflation. There were also particularly sharp rises in 1990 to 1992 and from 1998 to 2001. The annualized average growth rates of the arithmetic mean, geometric mean, and CPI-U are 10.8, 9.4, and 3.0%, respectively. As the correlation between the growth rates of geometric mean salary and arithmetic mean salary is over 0.90, normalizing relative to the algebraic mean does not meaningfully alter our results.
As our goal is to relate a player’s salary relative to the geometric mean salary of free agents, a natural starting point would be to simply calculate a simple logarithm of the ratio, ln(ratio) = ln(salary/gmsal), where gmsal is the league-wide geometric mean salary for free agents. To solve the mechanical problem of below-average salaries yielding negative ln(ratio) results, we simply add one and divide by a number, ln(100), which is sufficiently large such that transformed ratio is non-negative. Thus, lnnsal = 1 + ln(salary/gmsal)/ln(100). The function calibrates the statistic so that a player with the (geometric) mean salary has an lnnsal of 1.00, a player with double the mean salary will have lnnsal of ln(2)/ln(100) = 1.151, and a player with half the mean salary will have lnnsal of ln(0.5)/ln(100) = 0.849.
The unadjusted salary data is quite skewed (skewness = 2.94, kurtosis 15.21). The skewness of the transformed salary measure (lnnsal) is −0.08, while the kurtosis is 1.82.
The mean adjusted salary (lnnsal) for players with 0 years of experience is about 0.45. Inverting the formula, the simple ratio of salary to the geometric mean salary is exp((lnnsal − 1) * ln(100)). For lnnsal of 0.45, this equates to about 0.0794, or about 8% of the geometric mean salary.
The mean adjusted salaries conditional upon years of experience confirm that the trend over seasons 1–6 is rising steadily, not in discontinuous jumps as would be true if freedom to contract increased discretely after the third and sixth season. One reason for this comes from possible measurement error in our estimation of arbitration and free agent eligibility. Another would be that teams will offer some players multi-year contract extensions before the player attains eligibility status, and a portion of the player’s expected salary increase at that future date can then be collected at the time of the contract extension, consistent with the notion of goodwill in contracting.
The unadjusted mean (dollar) salary level falls after 10 years of experience for all quintiles except the lowest ability players (q1).
If one were to assume that the efficient salary for a player of position-specific mean ability is the geometric mean salary of all free agents, and that deviations of ability from the position-specific mean should be rewarded with log-linear increases in salary, then a ratio of 1.00 could be considered efficient. It is not at all clear to us that either of those conditions should necessarily hold.
See also footnote 7 where we contrast our approach to that of the traditional Scully approach. While it is true that the structural causes of changes in the ratio may go undetected with this method, the magnitude of the change in ratio will still be correctly measured. For example, suppose that the two main components of change in the ratio were increased freedom to negotiate contracts and migration of high ability players from small markets early in their careers to large markets mid-career. While we will remain unable to decompose our overall salary ratio increase into estimates for what proportion of the change in relative salaries is due to each effect, the fitted salary ratios should nonetheless correctly measure the sum total of all the partial effects.
The only significant departure from normality is the spike that occurs at the truncation in the left tail caused by the league minimum salary.
As mentioned previously, however, given the multiple manipulations of the two statistics being compared, we cannot claim that a ratio of 1.00 represents market efficiency, but only that relative player salary is increasing in the ratio.
The contingent mean ratios for the lowest quintile of players vary more during years 6 through 11 than the ratios for the other quintiles, due to increased standard error from small sample sizes. But as the ratios are only varying between 0.83 and 0.93, this says more about the stability of the ratios for the other quintiles than about erratic pay ratios for lower ability players.
Burgess and Marburger (1992) find salaries of arbitration-eligibles to be 80 to 90% higher than for “ineligibles” (players in their first 3 years), and Kahn (1993) found that salaries for arbitration and free agent eligibles were 35 to 50% higher than ineligibles, but the comparisons in both papers were damaged by data from the collusion era. Marburger (1994) finds salaries for average players in their first 3 years to range from 18 to 41% of those of free agents. Vrooman (1996) estimates that Tier 1 (ineligible) players receive about 28% of the salary of free agents. Using data from 2000 to 2004, Hakes and Sauer (2006) produce a model that estimates Tier 1 player salaries at 17 to 21% of free agent salaries.
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Acknowledgments
The authors wish to thank Glenn Knowles and session attendees at the March 2007 Midwestern Economics Association meetings for their helpful comments on an earlier draft of this paper, as well as several anonymous referees.
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Hakes, J.K., Turner, C. Pay, productivity and aging in Major League Baseball. J Prod Anal 35, 61–74 (2011). https://doi.org/10.1007/s11123-009-0152-8
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DOI: https://doi.org/10.1007/s11123-009-0152-8