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Friday, April 20, 2007

On the business of baseball

Forbes has released it's estimates of baseball team revenues, operating income, and book value. The data are not necessarily perfect, but they are the best data that us sports economists have to use.

A couple of interesting facts: Average book value of a professional baseball team was about $431 million. The average professional baseball team had revenues of $170 million and operating income (roughly profits) averaged $16.5 million.

I grabbed attendance to the home games and calculated average total revenue per attendee: $69.34. Here's a scatter plot of attendance against average price (essentially a demand curve if each team is thought to be located on a different point of the industry demand):

Average stadium capacity usage of professional baseball teams in 2006 was 69.34%. Since there are so many empty seats, why not drop prices a little lower and sell more tickets? This is an often-asked question, and the short answer is that teams likely find it profit maximizing to not sell-out every game. This seems counter-intuitive but is an important lesson from economics.

Having gathered attendance data, and overall team quality, I estimated a naive inverse demand curve using winning percentage as an instrument for season attendance:
. ivreg aveprice (avgatt=winpct06) 

Instrumental variables (2SLS) regression

Source | SS df MS Number of obs = 30
-------------+------------------------------ F( 1, 28) = 11.59
Model | 1977.39772 1 1977.39772 Prob > F = 0.0020
Residual | 2128.24154 28 76.0086263 R-squared = 0.4816
-------------+------------------------------ Adj R-squared = 0.4631
Total | 4105.63925 29 141.573767 Root MSE = 8.7183

aveprice | Coef. Std. Err. t P>|t| [95% Conf. Interval]
avgatt | -.0011032 .0003241 -3.40 0.002 -.001767 -.0004393
_cons | 104.1775 10.30607 10.11 0.000 83.06646 125.2885
Instrumented: avgatt
Instruments: winpct06

If we assume marginal cost of an additional ticket sold is zero (a rather benign assumption), then profit maximization would correspond with revenue maximization. Using the estimated inverse demand curve where quantity is measured as per-game attendance, the number of tickets that will maximize revenue is approximately 47,000:

. nlcom _b[_cons]/(-2*_b[avgatt])

_nl_1: _b[_cons]/(-2*_b[avgatt])

aveprice | Coef. Std. Err. t P>|t| [95% Conf. Interval]
_nl_1 | 47218.03 9284.486 5.09 0.000 28199.62 66236.43

If we look at the per-game attendance across the league in 2006 we find the following confidence interval:
. ci avgatt

Variable | Obs Mean Std. Err. [95% Conf. Interval]
avgatt | 30 31419.37 1609.978 28126.59 34712.14
This suggests that the average team sold 31,500 tickets per game and the 95% confidence interval was rather tight, between 28,000 and 34,700 tickets per game.

The revenue maximizing quantity obtained through the naive estimation is a bit higher than the actual number of tickets sold. However, if the slope parameter were -0.00165 rather than -0.0011, the revenue maximizing quantity would be 32,555 - almost exactly the number of tickets the average baseball team sells for an average game. It turns out that -0.00165 is within the 95% confidence interval of the slope parameter from the inverse demand curve. In other words, it is likely that firms are maximizing profits by maximizing revenues. Notice, as well, that the 95% confidence interval for the actual average attendance per game in MLB falls within the 95% confidence interval for the estimated number of tickets that correspond with profit maximization.

Another way to test this assumption is to test whether the price elasticity of demand is statistically different from -1. Estimating the direct demand curve in natural logs reveals the following:
. reg lnavgatt lnaveprice winpct06

Source | SS df MS Number of obs = 30
-------------+------------------------------ F( 2, 27) = 26.90
Model | 1.71411297 2 .857056486 Prob > F = 0.0000
Residual | .860294865 27 .031862773 R-squared = 0.6658
-------------+------------------------------ Adj R-squared = 0.6411
Total | 2.57440784 29 .088772684 Root MSE = .1785

lnavgatt | Coef. Std. Err. t P>|t| [95% Conf. Interval]
lnaveprice | -1.227604 .2343957 -5.24 0.000 -1.708544 -.7466637
winpct06 | .0012717 .0006013 2.11 0.044 .0000379 .0025054
_cons | 14.87012 1.162112 12.80 0.000 12.48566 17.25458
The hypothesis that the parameter on lnaveprice, which is an estimate of the price elasticity of demand, is different from -1 yields a t-test statistic of 0.94 and a p-value of 0.34. In other words, in the simple direct demand curve estimated it is not possible to distinguish the parameter estimate from -1.

What's the upshot of all of this? Sports economists have long asserted that professional sports franchises are profit maximizers (even if they provide some utility for their team owners) and, because the marginal costs are low, firms price and sell tickets consistent with revenue maximization which, in turn, maximizes profit. In this case, it seems that profit maximization occurs well short of the average stadium capacity (45,791).

By necessity, the profit maximizing number of tickets we estimate using a regression model is the average - some teams find it profitable to sell more tickets and other teams find it more profitable to sell fewer tickets. Moreover, there is nothing to suggest that all teams should make the same profits. Some teams will make more and some teams will make less than the average level of profits.

This often makes fans upset - their favorite team might not field a serious contender because the local fans are not willing and able to pay for a contender. In other words, those teams that are the best in the league year-in and year-out might have to be good in order to maximize profits. Still other teams maximize profits by qualifying for the playoffs once a decade and otherwise being an average or slightly less-than-average quality baseball team.

Here's an interesting but obvious tidbit: There is a high correlation between the ranking of a team's book value and the team's rank in annual attendance:
. corr bookvalue attend

| bookva~e attend
bookvalue | 1.0000
attend | 0.7840 1.0000

There is a positive but smaller correlation between book value and stadium capacity. Although the most valuable teams are located in the big cities, and therefore many people use the term "big market" teams to differentiate the "haves" from the "have nots" this distinction is not technically correct and might mainly exist for the benefit of disgruntled fans.

If we rank teams by their operating income (which is roughly equal to profit) we get the following:
| team profit~k operinc |
1. | Florida Marlins 1 4.30e+07 |
2. | Los Angeles Dodgers 2 2.75e+07 |
3. | Pittsburgh Pirates 3 2.53e+07 |
4. | Cleveland Indians 4 2.49e+07 |
5. | New York Mets 5 2.44e+07 |
6. | Colorado Rockies 6 2.39e+07 |
7. | Cincinnati Reds 7 2.24e+07 |
8. | Chicago Cubs 8 2.22e+07 |
9. | Seattle Mariners 9 2.15e+07 |
10. | Milwaukee Brewers 10 2.08e+07 |
11. | Tampa Bay Devil Rays 11 2.02e+07 |
12. | Chicago White Sox 12 1.95e+07 |
13. | Washington Nationals 13 1.95e+07 |
14. | Boston Red Sox 14 1.95e+07 |
15. | San Francisco Giants 15 1.85e+07 |
16. | Houston Astros 16 1.84e+07 |
17. | Baltimore Orioles 17 1.71e+07 |
18. | Atlanta Braves 18 1.48e+07 |
19. | Minnesota Twins 19 1.48e+07 |
20. | Oakland Athletics 20 1.45e+07 |
21. | St Louis Cardinals 21 1.40e+07 |
22. | Los Angeles Angels of Anaheim 22 1.15e+07 |
23. | Philadelphia Phillies 23 1.13e+07 |
24. | Texas Rangers 24 1.12e+07 |
25. | Toronto Blue Jays 25 1.10e+07 |
26. | Detroit Tigers 26 8700000 |
27. | Kansas City Royals 27 8400000 |
28. | Arizona Diamondbacks 28 6400000 |
29. | San Diego Padres 29 5200000 |
30. | New York Yankees 30 -2.52e+07 |

The New York Yankees lost money in 2006 (mainly because of the luxury tax), the San Diego Padres made a profit of 5.2 million, but the Florida Marlins (who had a sub-500 record) had a profit of $43m, even while griping about not having a new stadium. Here's something that might not be obvious to most fans: the correlation between attendance and operating income is -0.4199 and the correlation between winning percentage and operating income is -0.3328.

In the business of baseball, especially in an era of free-agent salaries and the luxury tax, the more the team wins, the lower the profits. What's going on? The source of this conundrum is the diminishing returns to quality on the revenue side: marginal improvements in team quality do not increase revenue as much. On the cost side, marginal improvements in team quality become ever more expensive.

Although many blame the "big market-small market" disparity for the reason why some franchises don't field a good team, the real reason is a "high attendance-low attendance" disparity and, more specifically, a "high willingness to pay for a contender- low willingness to pay for a contender" disparity. Those team owners who cite small market status as a reason for not fielding a contender might have a point - their market does not justify the money it would take to seriously contend for a championship. However, team owners often make it sound like they are taking it in the bottom line even while they can't build the fans a winning team. The data show a different story - lower quality teams in smaller markets to have higher profits than better teams in bigger cities (although for the four teams with the lowest payroll, the luxury tax helps pad the bottom line and steal from the teams with payrolls above about $148 million).

Stata Data File

Some work I did a while back implied that the franchise values imputed by Forbes (and it didn't really matterwhich sport one looked at) could be best approximated as

FV = c*(GR),

where GR is Forbes' estimate of a team's gross revenue. For MLB, at that time (about 3 years ago), c = 3.
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