Why was Ramirez suspended? Because he was using a banned substance, yes, but let’s unpack that. The purpose of the suspension is, presumably, to attempt to align the incentives such that a player who is tempted to use performance-enhancing drugs will find that the expected value of the marginal productivity of the drug use is lower than the expected value of the penalty. To break that down, there’s a probability π that a player who chooses to use banned substances will be detected, and a complementary probability (1-π) that he won’t be detected. As I discussed in an earlier post, we can run a regression and figure out what the values of the various statistics are worth. If the player is rational, he’ll be considering that using steroids will adjust his stats by some positive amount (i.e., that there will be a marginal product of drug use) and that will increase his salary when he next negotiates his contract. It’s also likely that the increased chance to be voted into the Hall of Fame or win a batting title, for example, will provide non-cash utility to the player, which we could also factor into MPdu. With the complementary probability, the player will be caught and will lose 50 games worth of salary (that is, the disincentive is 50*Salary/162, or 25/81 of his salary). Additionally, there will be disutility generated by the fans’ unwillingness to vote for him in the All-Star Game, for example, and the diminished likelihood of making the Hall of Fame.

If π*MPdu > (1-π)*25*Salary/81, then the player will rationally choose to use drugs.

If π*MPdu < (1-π)*25*Salary/81, then the player will rationally choose not to use drugs.

If π*MPdu = (1-π)*25*Salary/81, then the player will be indifferent between using drugs and not using drugs, so either choice makes sense based on the player’s tastes.

There are two main ways to decrease the proportion of players who use drugs – increase the probability of detection through more testing, or increase the disincentive to be caught using drugs by adding a lump-sum fine or increasing the length of the suspension (the 25 in our model) or both. I’m going to presume that 50 games was chosen as the length of the suspension for no good reason other than that it’s a nice big round number, and thus that the multiplier is essentially arbitrary. I’m also going to presume that Manny playing for the Isotopes imposes some positive externality on them and on the Dodgers – that the parent club will get better gate receipts from his appearances and that the players will benefit (probably by learning) from playing with a Major League-caliber player. The players in the Dodgers system will presumably be considered for MLB appearances at some point, and so the Dodgers benefit forM Ramirez‘s coaching function in his appearances at the AAA level.

(As a side note, a lump-sum fine would be a fine example of a Pigouvian Tax.)

It hardly seems fair that the Dodgers should benefit froM Ramirez‘s drug use. How do we solve this problem?

Auction Manny.

When a Major League player is suspended for drug use, allow him to make his 10 rehab starts. However, don’t automatically grant the right to assign him to the AAA club to the team he plays for. Instead, allow the clubs to bid on the right to have him do his rehab starts for their AAA team. Thus, Manny generates an externality on, say, the Buffalo Bisons (and therefore the Mets); however, the Mets have to pay an amount back to MLB that is, by the definition of an auction, more than anyone else was willing to pay.

A rational team will bid almost as much as they expect the player to generate in ticket sales and general utility, so the system is self-correcting with respect to the fame and ability of the player. However, MLB seems to benefit here. We can’t really allow that under a fair system, so I propose that the winning team’s bid be allocated to some combination of baseball development and drug education. Thus, every detected player loses $SuspensionMultiplier*Salary/162 in salary, some arbitrary club is granted the opportunity to profit from a shrewd bid but is unlikely to do so, and some combination of kids and drug education programs benefit about the amount that the arbitrary club feels Manny is worth to them as a AAA player.

Minor-league rehab
should not benefit users;
auction Manny off.

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On Friday, Luis Castillo committed an error in the bottom of the 9th inning with a one-run lead, two men on base, and two men out. The error was such that had Castillo made the play cleanly, the game would have ended with Francisco Rodriguez notching a save; however, Castillo’s error was directly responsible for two unearned runs scoring, giving Frankie a loss instead of a save.

The question: How much money does Castillo owe Rodriguez? I have a pretty good estimate.

Let’s assume, as usual, that clubs base their contract offers on the results the players create, that teams negotiate by placing roughly the same emphasis on the same statistics, and that the market for baseball players is competitive. It is then the case that when a player is a free agent, his stats will dictate the amount of money he’s offered in his next contract, and therefore we can model (using linear regression) the weight placed on each statistic. In this case, a linear regression model could tell us exactly how much money one save is worth come contract time.

Obviously, this is a complicated procedure, and it would take a lot of work to account for all possible variables, so to simplify matters, I did the following:

• Using Baseball-Reference.com‘s Play Index, I found all pitchers seven or more years into their careers (to avoid the inefficiencies of arbitration and rookies being locked into negotiating with their own teams and receiving the league minimum salary) who were free agents at the end of the 2007 season (the last free agency season available on Baseball Reference) and who pitched in relief in at least 80% of their appearances. There were 92.
• I threw out pitchers who did not play in 2008, to avoid distorting the salary output.
• I found the pitchers’ 2008 salaries and created a data file.
• Using that data, I ran a linear regression in R with these results.

First, the objections: there are obviously a number of vagaries in the data. We can’t, for example, easily account for popularity or clubhouse leadership, which are important factors in salary negotiations. This model has an R-squared statistic of .4174, meaning it explains 41% of the variation in salaries – not a huge number. The sample of pitchers is small, and it doesn’t account for people who might have had high reserve prices and refused to sign with any team despite having stats that would have helped solidify the model.

However, the SV (saves) statistic is highly statistically significant, and UR (unearned runs) is significant at the 90% level. Thus, we can estimate that if Frankie were to negotiate tomorrow for his contract, the figure that he was offered would be one save less and two unearned runs more than he would have been had Castillo not committed his error.

Since a save is worth $111,727, and an unearned run conceded is worth -$311,517, that would mean that Rodrigues would lose $[111727+2*(311517)] = $734,761 if he were to negotiate his contract tomorrow. Of course, he isn’t – he won’t negotiate until the end of the 2011 season. Thus, we have to discount twice.

Assuming a 20% interest rate, 734716*.8*.8 = 470,218.24 or about $470,000. (We’ll assume that teams discount the prior years’ performance and focus mainly on the immediately preceding year. 20% is a fairly high level of discounting, meaning that Frankie would be able to make up for lost stats by performing well next year and the year after.) Thus, Luis Castillo may have imposed a $470,000 externality on Francisco Rodriguez’ next contract negotiation.

If I were Luis, I’d offer him a steak dinner instead.

Luis Castillo
commits minor league error;
externality.