The root (Part 1)by Colin Wyers
February 26, 2009
It's a common question: How much is a baseball player worth?
And that's really the problem with passive voice: We avoid asking the really meaningful question here. A baseball player is worth very little to my aunt, for instance. A ballplayer is worth a great deal to me, but that fact isn't very meaningful, as I couldn't pay Ryan Theriot's salary, much less Alfonso Soriano's.
So let's limit ourselves to asking this question: how much should an owner be willing to pay a baseball player? This is probably a deeply unsatisfying question to be asking if you don't feel that baseball players should be making millions in the first place. But baseball itself is a business drawing in billions in revenue, and as the primary draws for the business are the players. It's not surprising that they expect to be paid in accordance with the amount of money being made from their labor.
So let's first ask how we determine the "right" wage for anyone.
Marginal revenue product, the short form
The economists (or some of them, at least) will tell you that the key to determining how much a person should be paid is their marginal revenue product. Before we go about trying to apply this to baseball players, let's try to see how it works in the real world.
Suppose you have a factory that makes sprockets. You want to know if you should hire more workers for your sprocket-making operation. You know how many sprockets people are willing to buy at different price levels. You know how much it costs to make a sprocket in raw materials. You know how many sprockets you can make for each additional unit of labor, and given the above information, you know how much additional profit those sprockets will bring in.
So, should Spacely Space Sprockets hire more workers, and how much should they pay them? The maximum they should pay them is the additional profit the company would earn from additional sprocket manufacturing—in other words, the additional revenue from additional sprocket sales, minus the costs involved in producing and selling those sprockets. The minimum they should pay them is probably whatever amount those workers could get for doing the same work over at Cogswell's Cogs. In a competitive labor market, these two figures should be pretty close to the same. If Cogswell is willing to offer $5 for labor that's worth $7, then Spacely should be willing to pay $6 for that labor to draw them away from Cogswell. After all, he's still making money at that higher price point. And then Cogswell should be willing to bid up the price of labor as well.
Practically, the "bidding up" process should stop before the maximum; after all, there's no point in bidding up for $0 additional revenue. But that could realistically be construed as the point at which someone is overpaid from their employer's point of view. Since the employer is the one that's paying the wages, I really don't know if I can come up with a more relevant point of view, so for now let's go with that.
MRP is typically expressed in the form:
Marginal Revenue Product = Marginal Revenue * Marginal Product
In other words, if each additional sprocket is worth $1 in revenue, and an additional hour of labor will produce 7 sprockets, the MRP is $7. This is gross MRP, or additional costs in materials, training, and so on required to add additional hours of labor are subtracted to provide the net MRP, which is what we're really after.
So how do we apply the principles of MRP to baseball?
Generally recognized as one of the earliest and of the most influential papers ever written on estimating how much a baseball player should be paid is Gerald Scully's "Pay And Performance in Major League Baseball." Published in 1974, we should recognize that many things have changed since then, both in the ways that baseball's finances and player compensation work and in the ways that we can measure a baseball player's performance on the field. So let's restrict ourselves to looking at Scully's model in broad strokes, not in tiny details.
Let's begin by defining what we mean as product in the context of baseball. Scully (and most subsequent salary analysis I've seen) has operated under the assumption that team wins are the product offered by major league teams. From a player evaluation point of view, this is probably as close as we're going to get, as it's difficult to price a player's other contributions to team revenues, and a player's contribution to team wins should capture their value to the team's revenue pretty well.
So Scully proposes the following model for figuring out a player's MRP:
- Figure out the marginal revenue per win for the average team.
- Figure out how many wins a player contributes to his team, based upon his individual performance.
- Multiply revenue per win by a player's win contribution.
Now, there are fundamentally two methods of rating a player's contributions to his team:
- Looking at a player's production as a proportion of the total team production. The critique of this method, from economist Andrew Zimbalist, is that this operates from an assumption of the alternative is a player who hits .000/.000/.000—essentially, the alternative is that the team forfeits those players at-bats.
- The incremental model, which looks at a player's production relative to average. Of course in the absence of a particularly crucial piece of information: the value of an average player. This really fails to tell us what a player's total worth is, so it is agnostic on the question of whether or not baseball players are over- or underpaid. This is something of a significant question.
This tension there should be familiar to anyone that read my player value series. If Spacely wants to produce more sprockets, he can simply hire more workers, or pay his current workers to work more hours. There can be costs to expansion that need to be considered, such as increasing plant space or running the plant for more hours, but there's no fixed limit to how many man-hours Spacely can employ in the manufacture of sprockets.
The same is not true of baseball. A baseball team has fixed limits, both of numbers of employees (25) and of time worked (outs). Consider a baseball team that is looking to increase its productivity. Suppose there are two players available in free agency, one who produces five wins for his team and one who produces three wins for his team (assuming a starter's playing time). Our sprocket-making factory would hire them both and pay them accordingly. Our baseball team may be only able to hire one of them, and they may have to displace a two-win player from the lineup in order to do it.
One critique of an MRP-based system is that it uses total revenues to estimate marginal revenues, typically using a regression model. We know there are several kinds of revenue, like broadcast revenues, that are weakly tied to single-season won-loss records; some, like merchandising and national broadcasting contracts, that aren't tied to win-loss records at all; and some, like revenue sharing, that are actually inversely related with win-loss records. This means that MRP-based salary systems can overestimate salaries.
Economist Anthony Krautmann has proposed an alternative model, called the "free market returns" approach. Essentially, estimates of the value of a given measure of production are measured against actual free-agent salaries, not revenue estimates.
This produces a model that can measure against real-world salaries much better than MRP-based estimates. Unfortunately, this assumes that MLB teams are paying the appropriate amount for labor; this is not always the case.
It should be noted that the Krautmann model is very similar (in broad strokes) to common sabermetric practices—most of the player salary estimates you see published from non-economists simply multiply their preferred player value metric by the average dollar value of that metric in salary.
Next week, we'll construct models of both salary approaches (or at least begin to do so).
References and Resources
For a look at the origins of baseball in relation to commerce, I heartily recommend "Their Hands Are All Out Playing."
Here's a look at a salary system based upon Win Shares. Baseball Prospectus uses MORP. Fangraphs also has a value-to-salary conversion system.
Andrew Zimbalist lays out his MRP model, and his critique of the Scully method, in "Baseball and Billions." Krautmann's critique comes from his paper "What's Wrong with Scully-Estimates of a Player's Marginal Revenue Product."
Special thanks to all in the Lounge - especially Jeff K. - who helped out with the article. "Spacely's Space Sprockets" is of course a reference to "The Jetsons."
Colin Wyers knows exactly how much of a nerd he is. He is very interested in hearing about any other concerns you may have; you can reach him by e-mail, and he will try his best to respond in a timely fashion. He also blogs at Statistically Speaking.