John Gobok ‘12
Harvard University, Department of Economics
This paper evaluates the impact of the National Basketball Association (NBA)’s soft salary cap and luxury tax system. First, analysis of standard deviation of team winning percentages from 1976 to 2010 shows that league parity worsened in general, but imposition of the cap in 1984 temporarily improved parity. Second, a regression of a team’s winning percentage on payroll deviation from the league median, while controlling for tax regime and team-fixed effects, shows that a positive relationship between a team’s payroll and performance exists. Third, there is a positive, significant relationship between a large-market team’s performance and the presence of the luxury tax. After the imposition of the tax, it is observed that the talent disparity between the middle- and large-market teams has increased. This counterproductive effect of the tax is explained by the discontinuity of the cost function at the tax threshold, effectively levying a higher implicit marginal tax on middle-market teams. By studying the mechanism of the Earned Income Tax Credit, the NBA can adopt a phase-out system for income subsidies, and incorporate the MLB’s gradation of tax rates into its own system.
Promoting league parity and competitive balance has always been a challenge to professional sports leagues. American sports teams are spread all over the country, with some teams located in large-market metropolitan areas while others are in small-market cities. As a result, there are considerable differences in terms of the financial resources available to teams, potentially resulting in league competitive imbalance as richer teams are better able to procure more expensive, and presumably, better talent. In James Quirk and Mohamed El-Hodiri’s (1974) seminal paper that modeled sports leagues, Quirk and El-Hodiri concluded that the rules of professional sports per se are relatively ineffective in balancing playing strengths, and attributed the franchises’ different drawing potential for talent as the main determinant of this imbalance. To address this, leagues have instituted a variety of measures—e.g., leagues have set limits on teams’ payrolls in the form of a salary cap, outwardly discouraged spending by progressive taxation through a luxury tax, and redistributed tax revenues to smaller-market teams.
This paper evaluates the impact of the National Basketball Association (NBA)’s soft salary cap and luxury tax system. The NBA salary cap is “soft” as it allows for certain exceptions for the cap to be exceeded. Its luxury tax system levies a dollar-for-dollar charge for any amount that exceeds a specified threshold, proceeds of which are distributed to nontax-paying teams. This paper investigates whether the current system achieves its stated aim of promoting competitiveness in the league. This paper adopts the standard definition of competitiveness used in sports literature, which is that teams have an equal chance of winning a randomly chosen game. Competitive balance is of primary importance to league sustainability insofar as a more competitive league boosts its quality and attractiveness to fans, leading to larger revenues and higher demand for its product (Ross 1997). Moreover, competitive balance ensures the survival of small-market teams. Based on an empirical study on various professional sports leagues conducted by Quirk (2004), small-market teams on the wrong end of the competitive imbalance suffer from decreased revenues which threaten their sustainability and, as an extension, the league. With the NBA scheduled to renegotiate the salary cap and luxury tax mechanisms in 2011 as part of the next Collective Bargaining Agreement (CBA), evaluating the current mechanism’s effectiveness in promoting competitiveness and financial sustainability for teams is especially important.
This paper employs several methodologies for examining the effectiveness of the cap-and-tax system of the NBA. For the approaches outlined in this paper, the years 1976 to 2010 are divided into three time periods: pre-cap (1976-1983), soft-cap (1984-2000), and cap-and-tax (2001-2010). First, the ideal and actual standard deviations of team winning percentages were compared to see the general trend in league competitiveness during the three regimes. Overall, league competitive balance declined, with the soft-cap era being the most imbalanced. The introduction of new regimes has the temporary effect of improving competitive balance immediately after they are instituted, but effects seem to wear off in the later years. Second, a regression of winning percentage on team payroll was run to capture the relationship between payroll and talent, under team-fixed effects. The results indicate that there is a significant, positive relationship between talent and payroll. Furthermore, the imposition of the luxury tax also has a significant and positive impact only on the winning percentages of the richest teams. Next, the cost function discontinuity due to the luxury tax is analyzed by comparing it to the implicit marginal tax rate workers face due to the combination of progressive taxation and social insurance benefits. To check empirically the initial impact of the tax, year over year change in payroll before the tax is modeled as a function of the previous year’s payroll, team performance and market size. Using this model, 2001 team payrolls are predicted, which is then compared to their actual 2001 figures. Findings show that the tax hit the highest spending teams the most, at least initially. To gauge the long-term impact of the tax on different teams, teams were tiered into three groups depending on their payroll levels. A graph of medians of the three groups shows winning percentages between the highest-spending teams and the middle-spending teams widening as one moves away from the year of implementation. Lastly, from these analyses and findings, potential recommendations are discussed in order to better achieve the system’s aim of promoting competitive balance, like phasing-out of tax payouts and the gradation of tax rates.
The NBA soft cap and luxury tax mode: League overview
The modern NBA was created as a result of the merger between the NBA and the American Basketball Association in 1974. Starting with 22 teams based in various US cities, today it has 30 teams, 29 in the Continental US and one in Toronto. Teams derive their revenues from three sources: (1) 1/30 share of league-wide revenue from merchandise, national and overseas TV contracts; (2) local TV contracts, sponsorships and gate revenues; and, (3) luxury tax payouts, if eligible. Since teams are located in different areas with varying market sizes, their total revenue can differ significantly due to more lucrative TV contracts, sponsorships and higher ticket sales in metropolitan areas. For example, in 2009 the New York Knicks had a revenue of $202 million while the Memphis Grizzlies earned $98 million, less than half that of the Knicks. Economic forces dictate that richer teams can afford to pay more expensive and talented superstars, leading to potentially lopsided games. While the league has grown in terms of size and revenue, this challenge of maintaining competitive balance is key if the league wants to continue fostering fan interest and ensuring financial sustainability of the teams (Noll 1974).
The soft cap
In response to the potential arms race breaking out between the richest teams, the NBA instituted a salary cap in 1984 to put a ceiling on team payrolls. Currently, the cap is set at 57% of projected Basketball Revenue Income (BRI), which is forecasted in the CBA and adjusted accordingly with unforeseen year-on-year economic changes. Fort and Quirk (1995) modeled how a salary cap can increase competitive balance in the league while also ensuring the financial stability for teams in weak-drawing markets. In line with this, the NBA initially proposed to have a “hard” cap like the NFL, where teams cannot exceed the cap for any reason. However, the hard cap proposal was completely unacceptable to the NBA Players Association, arguing that it would result in players being paid grossly below their market value. With the cap negotiations being part of the CBA, the relatively stronger players’ union in the NBA (compared to that of the NFL) makes the hard cap proposal a tough sell. Thus, the NBA relaxed this cap proposal by providing a family of exceptions, ultimately leading to the NBA’s “soft” cap. Among these exceptions are allowing teams to retain their own players who have played for them for the past three consecutive years, sign veterans to a contract equal to the league average, roster rookies in accordance with the rookie salary scale, even if these measures lead teams to exceed the initial cap. In practice though, as it has been observed that since its institution, teams who intend to go over the cap very conveniently utilize these exemptions. The data set used in this paper show that at least 80% of teams exceeded this soft cap in the years that it was implemented. Nonetheless, an important consequence of the soft cap is the limit it places on player movement as transaction salaries also needed to match in order for trades to go through (Banks 1997). By itself, the soft salary cap was ineffective in improving competitive balance due to its permitted exceptions, which were easily utilized by teams at will. With this in mind, the NBA added another feature to its salary-controlling mechanism in the form of the luxury tax.
In recognition of the limitations of the soft cap’s effectiveness, the NBA instituted the luxury tax starting the 2000-2001 season. The concept of the luxury tax came from the MLB, which does not having a salary cap. Teams are subject to a luxury tax when they cross a certain payroll threshold and proceeds are distributed to nontax-paying teams. In theory, this would provide a disincentive to high-spending teams, decrease their expenditure and spread talent more evenly between teams. Instead of setting a hard cap, the NBA chose to spread the benefits that large-market teams have to other teams in the league (Vrooman 1995). This is similar to how most governments use the tariff system over the quota system in regulating imports. A tariff system is able to generate extra revenue, which can be redistributed to other economic agents affected by the excessive imports. The NBA adopted this model by charging a dollar-for-dollar tax for amounts in excess of a stipulated threshold, currently set at 61% of BRI. Under the initial luxury tax system stipulated in the 1999 CBA, a tax only kicks in if the league payroll is collectively higher than 61% of league-wide BRI. Hence, even if a team has a payroll higher than 61% of the average team BRI, it may not need to pay taxes so long as the total league payroll is less than 61% of league-wide BRI. Such was the case in 2002, where there were teams that crossed the threshold, yet collectively the league was under it, hence no team paid any tax. This uncertainty was removed in the latest CBA, in which, from 2006 onwards, the luxury tax was certain to set in if a team’s payroll crossed 61% of the NBA average team’s projected BRI. The tax revenues are divided by 30, with each non tax-paying team getting a 1/30 share of the tax payout, thereby serving as an incentive for teams to stay below the tax threshold. The remaining amount is used for league purposes like operations and overseas marketing.
Trend in league winning percentage standard deviation
In investigating the impact of the regime changes on league competitiveness, the overall change in league competitiveness is charted based on the standard deviation of the team winning percentages for a particular season. One can compare the winning percentages’ actual standard deviation (ASD) with the ideal standard deviation (ISD) in a given year, assuming a perfectly competitive league. This approach was first used by Roger Noll in 1988 and popularized by Rodney Fort in 2007, after which it became the standard approach to measuring league competitiveness (Trandel and Maxcy 2009). Both the ASD and the ISD for the team winning percentages are calculated for each year between 1976-2010. In a perfectly competitive league, the probability of any team winning (or losing) any given game is given by the equation P(W) = P(L) = 1 – P(W) = 0.5. Thus, for an 82-game season, ISD = 0.5 / √82. The ratio of standard deviations (RSD) is calculated by finding the ratio of ASD to ISD for a given year (RSD = ASD / ISD). A larger value for RSD signifies a wider spread of outcomes, adjusted to control for the dispersion that could be expected from random variation, and thus a greater degree of competitive imbalance. The RSD for each season is plotted in Figure 1, with a season represented by a year that is the later calendar year (e.g. 2009-2010 as ‘2010’). The green lines on the graph (year = 1984 and year = 2001) mark the shifts between regimes, and the dashed red line (year = 2006) demarcates the adoption of the newest version of the luxury tax.
In general, between 1976-2010, RSD diverges away from 1, signifying that the league is becoming more unequal. From the graph, it appears that the imposition of the soft cap in 1984 initially improved league competitiveness, as RSD seemed to revert back towards 1. Nevertheless, between 1985 to 2000, league competitiveness went back to its original state and even worsened in the mid-90s. Immediately after the cap’s implementation, teams could have been retrained in their spending, but later on found loopholes and took advantage of exceptions to circumvent the cap, rendering it practically useless (Kaplan 2004). The exceptionally high competitive imbalance in the league in the 1990s can be attributed to the Jordan-era Bulls, where between 1996-1998 the team achieved the two winningest records in NBA history. During the 1997-98 season when the Bulls had won the previous two years’ championships, attendance in smaller-market cities were down 20% from recent historical averages and nearly half of franchises were losing money (Leeds and von Allmen 2002). No recession hit the economy that year, so analysts underscore the huge competitive imbalance as the reason for declining gate revenues. Chicago was able to build and maintain a dynasty largely with the help of the soft-cap. The cap limited league player movement and allowed the Bulls to re-sign its players at much higher salaries than other teams could offer due to the fact that they have already played for the team for more than three years. This could be extended to other dominant teams in the 1990s like the Utah Jazz, Los Angeles Lakers and Detroit Pistons, who were able to retain their own high-paid superstars with the help of the soft-cap exceptions that advantaged them in contract extensions.
During the luxury tax era, competitive balance seemed to have improved as RSD shows a general decrease from 2001-2007. What happened in the later part of the decade runs opposite to its earlier half, although it is still noteworthy that the average RSD is better than it was in the 1990s. Hence, Figure 1 hints at how the salary cap contributed to the building of dynasties and the ensuing greater competitive imbalance, while the impact of the luxury tax is less certain.
Relationship between winning percentage and payroll
This paper assumes that paying higher leads to better teams. This assumption is tested in this regression of winning percentage, win, on payDev, defined as the ratio between a team’s payroll and the median payroll of the league in that particular year. The data used is from 1986 to 2010, excluding 1987 and 1989 due to insufficient information. The dummies regime, taxMid, and taxBig are added to the regression to see the impact of the presence of the luxury tax on win, and the interaction variables are there to measure the tax’s impact on middle- and big-market teams (assuming market size is proportional to their payrolls). Team-fixed effects are also added to control for team-specific characteristics. Relevant results are displayed in Table 1.
From Table 1, the coefficient of payDev is 0.174, which is significant at the 5% level. Assuming all else constant (including controlling for the regime and the team), a unit increase in payDev (increase in a team’s payroll equivalent to the league median) increases that team’s winning percentage by 0.174 or adds about an extra 14 wins. In a league where a win of 0.50 can almost certainly guarantee a playoff spot, this is significant. Hence, these regression results support the assertion that there is a positive correlation between a team’s payroll and talent.
The coefficients of regime and taxMid are not significant at the 5% level. There could be no observable impact of the presence or absence of the tax on the winning percentage of a team. The same is true for the tax’s impact on mid-market teams, where there is no significant impact. The interesting result here is the 0.042 coefficient on taxBig, which is significant at the 5% level. All other things constant, the imposition of the luxury tax has a positive impact on big-market teams’ winning percentage, adding about 3.5 wins (0.042 × 82). While the tax was set out primarily to limit spending of large-market teams in the hope that they do not become too good, the regression tells us otherwise. The positive and significant impact of the tax on the large market teams is something that succeeding sections will further analyze.
Since team effects are deemed to be more significant than year effects, the team-fixed effects are used in the said regression. While year-to-year talent availability in the league may vary, superstars are usually signed to long-term contracts, hence team-fixed effects become more significant as talent availability does not change much from one year to another. Team-fixed effects can control for coaching staff quality (which is not subject to the cap), training facility quality (larger markets may have better stadiums and professional trainers), marketability to superstar free agents (big-city appeal) and fan attendance (in-game cheering). With that said, a year-fixed effects regression is also run, and the results are comparable. The coefficient of payDev is 0.169, and the taxBig is 0.0725, which is significantly higher than that in the team-fixed effects model. The team-fixed effects model does a better job of explaining the variations in win, being able to explain 26.5% of variations in winning percentages compared to 10% for the year-fixed effects model (adjusted r2 of 0.265 vs. 0.101).
It is important to note though that the model itself may not accurately predict the win percentages of teams since adjusted r2 is only 0.265. The constant term 0.341 is still very large; by itself, it signifies that just the mere presence of any team in the league would give it a 0.341 win. Nonetheless, the coefficients for the team dummies provide strong evidence that smaller market teams are inherently disadvantaged in competing in the league. Table 2 lists the team coefficients that are significant at the 5% level, along with their market size ranking out of 30 NBA teams.
From Table 2, it can be deduced that most of the teams that are negatively affected by the team-specific characteristics are usually small-market teams. The Phoenix Suns, 15th out of 30 in terms of market size, is dropped in the regression and used as the point of reference for the coefficients. The negative coefficients are interpreted as the drop in a team’s winning percentage solely due to the team’s inherent characteristics. For example, Indiana has a coefficient of -0.130, which means that its winning percentage, keeping all other things constant, decreases by 0.130 simply because it is the Indiana Pacers. Out of the eight teams with significant negative team coefficients, six of them belong to the bottom half of the league in terms of market size. This is intuitive since smaller market teams like Indiana, Charlotte and Utah earn less local revenue which could be used to hire better coaching staff, secure state-of-the-art facilities, and fund expert scouts. Furthermore, bigger markets can be a draw to superstar players; this was the main reason why Lebron James entertained leaving Cleveland for cellar-dwellers New York, New Jersey and Chicago during the summer of 2010 (O’Connor 2010). Hence, these intuitions and the regression results support the notion that smaller market teams are inherently disadvantaged in terms of competing in the league.
Counter-productive Luxury Tax Model
For the purpose of analyzing the tax’s effects on different types of teams, teams are classified under three categories A, B, and C. Group A comprises teams that are already over the tax threshold (between 1986-2000 when the tax did not exist, tax threshold is assumed to be 61% of BRI). Group B comprises teams under the tax threshold but over the soft cap (teams with payrolls between 57-61% of BRI). Group C comprises teams under the cap.
To see the initial impact of the tax on team payroll, a model of predicted team payroll is constructed for the period before the tax implementation. In this regression model, current team payDev is regressed on the previous year’s payDev and win, while also including a team’s group classification from the previous year assuming that market size is positively correlated with payroll (llow, lmiddle, lhigh). This model used data from 1996 to 2000, a period when the league size and composition remained. After obtaining the model, 2001 team payDev are predicted and compared to the teams’ actual 2001 payDev. Figure 2 plots this difference between the predicted and actual pay deviations against the team’s 2000 payDev, which is assumed to be indicative of market size. The graph shows how the tax affected the payroll decisions of teams of different market sizes.
From the Figure 2 plot, the difference for teams with 2000 payDev greater than one is mostly positive, indicating that the predicted payDev was higher than the actual payDev; hence, the tax moved the large-market teams’ payrolls closer to the league median. For teams with 2000 payDev less than one, the difference is mostly negative, indicating that the tax increased their payDev and moved it closer to the league median. As a result, the spread between the payrolls of the highest- and lowest-paying teams has narrowed with the tax. Such a movement towards the median for both extreme groups shows how the tax decreased the league payroll deviations, which was the original intention of the tax.
However, from the graph of ratio of actual and ideal standard deviations (Figure 1), league competitive imbalance seemed to increase a few years after the tax implementation. As such, evaluating the impact of the tax warrants examining its long-term impact on the teams. Critiquing the first system of the NBA Luxury Tax, Kaplan (2004) asserted how the system was misguided as it worked against its stated goals of promoting competitiveness. In a sample case that outlined implicit marginal tax rates for teams in the respective groups A, B, C, he showed how Group B teams are implicitly taxed the highest, which puts them at a disadvantage against Group A teams. This paper utilizes the same approach of analyzing the impact on the three groups under the new system, and also adds empirical analysis of the teams’ winning percentages to support the claim that Group B teams are indeed disadvantaged even with the modified version of the tax regime under the 2005 CBA.
To investigate the long-term impact of the tax on Groups A, B, and C, the median winning percentage is calculated for each group from 1986 to 2010. Due to the small population size of teams under Group C, analysis is focused on the difference between the medians of Groups A and B for each year. This difference in medians is assumed to reflect the talent gap between Groups A and B. Figure 3 illustrates the median winning percentages for each group as well as the difference between median values of Groups A and B between 1986-2010.
In Figure 3, the spread between median team winning percentages of Groups A and B decreased right after the imposition of the luxury tax in 2001, with the difference drastically dropping to about 0.1 from 0.55. This resonates with the finding in Figure 2, where the team salaries moved closer to the league median immediately after tax imposition. As such, the introduction of the tax decreased both payroll deviations and talent disparity in the league. The regime change could have compelled the teams to rethink their payroll decisions, making high-spending teams averse to spending more.
However, this initial effect is short-lived, as indicated by the post-2001 upward trending line of the difference between medians in Figure 3. People tend to overreact to regime changes, and after a few years the highest spending teams could have reverted to their old ways since marginal benefit of spending more (more wins and advertisements) is higher than the marginal cost (even after incorporating tax).
The increasing talent disparity between Groups A and B can be attributed to how the current luxury tax system causes teams that cross the threshold to fall off a “cliff.” On one hand, crossing the tax threshold forces them to pay a dollar-for-dollar tax for any amount in excess. Furthermore, it also disqualifies them from receiving tax payouts, which can be substantial for a small- and middle-market team. In this context, implicit marginal tax rate is defined as the combination of the luxury tax payments and the effective loss of tax payout eligibility. Table 3 lists the total tax revenues and the payouts to each non tax-paying team during the latest CBA.
When determining the optimal payroll amount, teams face a similar problem to what workers face with the presence of welfare benefits and a progressive tax regime. In the real world case for workers, the middle-income earners face a higher implicit marginal tax rate than the highest income earners due to the middle incomers’ loss of eligibility to social insurance programs. This is analogous to the NBA luxury tax model, where teams just at the threshold face a considerably higher implicit marginal tax compared to teams already above the threshold since crossing the threshold implicitly taxes them extra by making them ineligible for tax payouts. Figure 4 illustrates the discontinuity in the cost function that a team in Group B faces.
Figure 4 displays the cost function that firms face when deciding to hire additional talent. For Group C, marginal cost is just equal to the cost of the talent, while for Group A, it is twice the cost of the player’s salary due to the 100% marginal tax rate. For Group B, the marginal cost of hiring talent is the highest among the three groups due to the discontinuity at the tax threshold. Thus, a Group B team will be less likely to hire that additional talent. To illustrate this, assume that a $10 million superstar is on the market, and the cap and tax thresholds are $50 million and $60 million, respectively. Tax revenue is assumed to be at $120 million, the 2010 level. The cost of signing this player is $10 million for Group C, $20 million for Group A, and $24 million for Group B. These cost calculations are detailed in Table 4.
As teams in Group B face the highest cost, this new tax system hits them the worst and may even exacerbate the talent disparity between Group A and Group B teams. This can plausibly explain the increasing difference between the medians of Groups A and B after the tax was imposed. Thus, the discontinuity in the cost function facing Group B teams makes the tax system counter-productive in promoting talent parity between large-, middle- and small-market teams.
The previous sections show how competitive balance has worsened in the NBA since its inception, and how the current luxury tax regime, due to the discontinuity of the cost function at the tax threshold, is counter-productive in improving talent parity in the league. Examining the Federal Government’s Earned Income Tax Credit (EITC) system and the MLB’s luxury tax model offer some insights as to how to make the NBA system more effective in promoting competitive balance.
The EITC was enacted in 1975 to provide a subsidy for low-income working families. The credit equals a fixed percentage of earnings from the first dollar of earnings until the credit reaches a maximum. The credit then stays flat at that maximum as earnings continue to rise, but eventually earnings reach a phase-out range. From that point the credit falls with each additional dollar of income until it disappears entirely. The three phases of the 2010 EITC for various family types is shown in Figure 5.
The NBA Luxury Tax payout system can adopt a similar approach, specifically a phase-out system, to that which the EITC employs. To ensure that the implicit marginal tax rate increases as one moves from Group C to A, teams should not be automatically disqualified from receiving tax payouts once they cross the tax threshold. Instead, payouts should gradually be decreased until it eventually becomes zero. This also allows more flexibility to Group B teams to exceed the threshold by a small percentage. For example, for teams with payrolls between 61-65% of BRI, they still remain eligible to tax payouts but at a decreasing percentage inside this phase-out range. By doing this, it increases the total tax payouts, and given that each eligible team always receives 1/30 of the total payouts, low-spending teams receive even a larger amount than what it would have under the current system. Such a tax regime still penalizes big-spending teams but also redistributes more money to less advantaged teams—money that these less affluent teams could use to improve their roster or secure better staff.
Moreover, the tax penalties could also be graded based on how much teams have exceeded the threshold and their frequency of exceeding the threshold for the past five years—a gradation scheme similar to that of the MLB. In the first type of gradation scheme, based on the amount that the team exceeds the threshold, teams can be charged a 100% marginal tax rate on the excess amount if payrolls fall between 61-65% of BRI, and then a 125% tax if payrolls exceed 65% of BRI. Combining this with the phase-out system of payout benefits ensures that Group A teams will have the highest implicit marginal tax rates among the three groups, potentially narrowing the talent gap between Groups A and B.
Furthermore, in the MLB’s tax model, frequent offenders pay a much higher tax rate as compared to first offenders: 17.5% for 1st offenders, 30% for 2nd offenders, and 40% for succeeding offenders. In a way, this ensures that the larger market teams, most probably the most frequent offenders, are taxed more than the middle-market teams who just happen to cross the threshold this year. This idea of gradation based on frequency of offence could be incorporated into the NBA’s tax system.
In summary, this paper calls for the creation of a phase-out stage for tax payouts, and the gradation of tax penalties based on the amount in excess and the frequency of such an offence in the past five years. Implementing these policies would remove the discontinuity that Group B teams face and would make the curve much steeper as one moves further away from the threshold. Figure 6 illustrates the cost functions facing the three Groups under this proposed system.
The challenge of creating a more competitively balanced league has always been a perennial one facing market designers. The NBA initially adopted a soft salary cap, which only aided the building of dynasties in the 1990s. It then introduced a taxation and redistribution mechanism in the form of a luxury tax in the hope of reigning in the spending of the large-market teams and subsidizing the small-market teams. However, empirical results have shown that the tax has positively impacted the winning percentages of the highest spending teams, suggesting a widening of talent disparity between the high and middle spenders. This counterproductive effect of the tax is explained by the discontinuity of the cost function at the tax threshold, effectively levying a higher implicit marginal tax on middle-market teams. Such discontinuity is a common problem in many markets; thus, the NBA can adopt social insurance’s phase-out system for income subsidies, and incorporate the MLB’s gradation of tax rates into its own system. With the current CBA set to expire next year, these findings should be taken into consideration when structuring the regulatory mechanism going forward. Nonetheless, it is important to note that this system is negotiated collectively under the whole CBA, where competing interests may trump economic sense.
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 Removing the Bulls (and other dynasties) from the analysis would not lead to a significant difference in results. Their impact on the analysis is mostly on the winning percentages of other teams, as their exceptional talent causes the league-wide standard deviation to increase.