Explaining Spending: Deriving Teams’ Optimal Spending Strategies

A couple weeks ago, I made the argument that the cost of a win on the free agent market for a given season does not represent a generally homogeneous league-wide conception of what a win is worth. Rather, the more logical explanation is that it reflects the largest amount any team would be willing to pay for the Nth win available after the first N – 1 wins were hypothetically distributed in accordance with who valued each of them most highly. (Economic theory can sound weird when you apply it to real life.)

As an economics student and baseball fan I find this idea to be fascinating, but I admit that the origin of the price of a win doesn’t have many practical uses in concrete baseball analysis. However, if I am correct, this theory does have two important implications for understanding how MLB teams (should) operate: that the price of a win and the value of a win are two different things, and that this uncoupling means a marginal win can be worth significantly different amounts to different teams.

With these assumptions in our minds, it is possible to construct a reasonably simple model for a team’s optimal player-signing and front office employee-hiring strategy in the league labor market.

The Perfect-Information and -Competition Model
Whenever you consider buying a good or service, the first question you should always ask yourself is: How much enjoyment or need-fulfillment will I get from it? For those who have never taken an economics class, this concept is called “utility,” and in any given situation a rational actor’s goal is to maximize it. Every agent’s utility function is different and based on different factors; an MLB organization’s utility is probably defined primarily by how much money it makes and how well its major-league team is playing.

Let us use A_t to represent the total value of a potential employee (player or otherwise) to his or her team t in terms of the team’s own utility function. I posit that this can be hypothetically calculated by:

where U_t is team t’s utility as a function of games won and expressed in dollars, w’ is the number of games the team would win if it signed or hired the player or employee in question, w is the number of games the team would win without him or her, and J_t is the off-field value of the employee to the team.

What does that actually mean? Let’s start by looking at the most basic variable in the equation: wins. At a fundamental level, every team’s long-term goal is to win as many games as possible, and every move a team makes is (or at least should be) with some willful intention of winning more games. You can think of a player or front office staffer’s contribution to his or her team’s win totals (i.e., w’ – w) in any terms you want—WARP, pitchers’ wins minus losses, RBI divided by caught stealings, whatever—but the unit of desired outcomes is universal.

Next is utility as a function of wins (U_t). For every number of games won, a team and its owner(s) will derive some level of happiness. The value of a marginal win depends on where the team currently is on the win curve—going from 60 to 70 wins probably isn’t as valuable as going from 85 to 95—but it is hard to imagine that it’s ever not positive. Winning more games leads to happier people in the office, including the ones holding the purse strings. It also leads to happier fans. Increased fan happiness leads to higher attendance, and higher attendance leads to greater revenues. The difference between a team’s utility at w’ wins and its utility at w wins (the combination of monetary and nonmonetary factors) should in most cases represent how team t values the employee’s contribution.

The exceptions to this are represented by J_t. This represents the value (monetary or otherwise) that team t receives from the employee in question beyond his or her direct contribution to the team’s on-field performance. It is designated as such in honor of Derek Jeter, who is the poster boy for this effect’s impact on teams’ behavior. The three-year, $51 million free agent contract Jeter signed with the Yankees after the 2010 season was far more than what another identically talented shortstop would have gotten from some other team in that market, but the combination of the revenue from fans coming to see Jeter and the enjoyment the Yankees have gotten from keeping him in pinstripes for life presumably had substantial value to the team. It takes an exceptional player-team combination for J_t to have significant value (I cannot think of a non-player employee who would have such an effect), but it truly can matter in certain cases.

Let’s say that the cost of signing or hiring the player or employee is C. Team t will bring him or her aboard only if it values him or her at least as much as he or she will cost—i.e., A_t ≥ C. You don’t need an economics textbook to tell you that.

However, there is another factor that should shape teams’ hiring strategies: opportunity cost. The economic model I described for the market for free agent wins implies that teams are “price-takers” in the market. This means exactly what it sounds like: once the equilibrium price is set it should be universal across the league, and no consumer should pay more (or be able to pay less) than that. The logic behind this is simple. Every dollar spent on one free agent is a dollar the team isn’t spending on other free agents. And if that dollar is buying fewer wins than the team would expect to be able to purchase with it at the market price, it should either lower its offer or go find someone else to sign.

We can express this mathematically, too. Given an established market price per win P_y in year y, the maximum amount B_t that a team should be willing to spend on an employee can be given by:

This formula should be fairly self-explanatory. In general, the most a team should be willing to pay to go from w wins to w’ wins is equal to the market price of going from w wins to w’ wins—otherwise, it’s overpaying. In special cases (like Jeter’s), a player comes with enough externalities to raise his market price above what his production would warrant, but again, that probably does not apply to most potential employees.

The concern over opportunity cost adds another restriction to teams’ optimal hiring strategies: it is worth bringing an employee on board if and only if both A_t ≥ C and B_t ≥ C. In a rational, perfectly competitive market with perfect information and general homogeneity of employee value across the league, only the former condition would matter; B_t could be assumed to equal C because the invisible hand would root out and smooth over any inconsistencies in the market. But of course, that’s not how things work in the real world.

Heterogeneity and Inefficiency
If we tug at the thread for the theory that B_t should equal C, we can see how the perfect-world model doesn’t translate well to reality. Specifically, I identify three reasons why an employee’s market value to a specific team could be lower than what he or she would actually cost: the heterogeneity of on-field value, the heterogeneity of off-field value, and the existence of inefficiencies in the MLB labor market.

First is the notion that not every employee has the same on-field value to every team. Imagine if Andrelton Simmons were a free agent and a strong gravitational field had spontaneously appeared at Fenway Park just to the left of second base. If almost every batted ball that Boston pitchers gave up started going towards the shortstop, then a strong defensive shortstop would be worth substantially more to the Red Sox than he would to any other team. Say Simmons would be worth somewhere in the neighborhood of four wins a year to the 29 other teams but that he’d be worth 20 to Boston. (I’m making these numbers up.) The Red Sox wouldn’t need to pay him as though he’s worth 20 wins if no one else is willing to give more than the market price of four wins.

We might see this process play out for non-player employees, too. Imagine that two teams are each looking for a new pitching coach at the same time. One team has a starting rotation full of in-their-prime Cy Young-caliber arms, while the other staff is composed entirely of prospects with unfulfilled potential and veteran reclamation projects. The coach might be worth only one win to the All-Star team because he wouldn’t have much to teach them, but perhaps his guidance would be worth five wins to the team that needs him more. The latter team would presumably sign him, but it would need to give the coach only marginally more than his other suitor is offering to bring him aboard.

There is probably a similar heterogeneity in players’ potential off-field values. Consider the case of Jeter. Any team probably would have sold some extra tickets had it signed a future Hall-of-Famer, and if his leadership skills and knowledge of how to win are worth even a fraction of what they are purported to be, he’d have a positive effect on any clubhouse he’s a part of. But unless the Red Sox were feeling particularly vindictive, his off-field value would almost certainly be worth more to the Yankees than it would to any other team. It didn’t work out this way empirically, but had they signed him at just above whatever the second-best contract offer he got was, his market price would have been less than the market equivalent of his value to the Yankees.

This leaves the last and most interesting reason why the market price of a win does not always equal the market price of a win: the existence of market inefficiencies. If the market is systematically undervaluing a certain type of player or skill set or front office employee, a team that realizes that their true worth can buy these cheaper wins at discount prices until the rest of the league catches on. The Moneyball A’s, for example, realized that the market prices of players with good plate discipline like Scott Hatteberg and David Justice were significantly lower than their market values.

That such an inefficiency can exist is a violation of the earlier assumption of perfect competition and also implies that teams don’t actually have perfect information about how much their players and employees will be worth, as also assumed earlier. Obviously, this is how real life works—the assumptions of perfect competition and information in the MLB labor market apply only in theory—which means we need a less-simplistic model for teams’ optimal hiring strategies.

The Imperfect-Information and -Competition Model
The models I described earlier are all well and good, but their practical utility is limited by what The West Wing Vice President John Hoynes would call “the sheer tonnage” of what teams don’t know. How many wins the team will win with its current roster, how many more games it will win if it signs a player or hires an employee, how much added revenue that will bring in, what the price of a win in a year’s market will be—a team can make good educated guesses for all of these variables, but they’re still just educated guesses. So we need a model that’s based on educated guesses.

Going back to the basic question of how much utility a player or employee would bring to a team (as previously represented by A_t), the expected in-a-vacuum value of a new signing or hire α_t to team t is given by:

where every variable is replaced by its expected value. The better a team is at both assessing the quality of its preexisting roster and projecting the future value of its potential acquisitions, the more likely it is to add employees at ultimately favorable salaries.

We can do the same thing to change the market value of a potential signing’s production or employee’s labor B_t into its expected value β_t:

where teams’ maximum bids for players and employees are defined not only by the expected values of their productions but by the expected price of a win. The former is generally of greater focus and import, but the latter is significant too. The 30 MLB ownership groups don’t come together to hypothetically auction off all N wins available in a given year so that everyone knows the equilibrium price. And misjudging the price of a win can lead to teams making deals that end up looking bad very quickly.

So a team’s decision to sign a player or hire a staffer is a function of his or her salary relative to both his or her expected utility-based value and his or her expected market value. Or, in other words, a team should pull the trigger on bringing an employee aboard at cost C if and only if both α_t ≥ C and β_t ≥ C.

I don’t think I’m blowing anyone’s mind by saying that teams should sign players and hire staffers only if the utility they get from their employees’ production is worth their salaries, and it doesn’t take an economics professor to tell you that you should never willfully spend more money than you have to. But as I’ve learned from years of economics classes, the best models are the ones that make you feel like the conclusion was intuitive.

MLB teams don’t always make the right decisions about whom to sign or hire. But they always think they are making worthwhile investments. And in trying to understand why a team signs a free agent or hires an employee, appreciating the relationships among the utility-based value of a win, the market value of a win, and the price of a win is essential to seeing their point of view.