There is a pretty accepted methodology to answer this question that actually predates free agency. First, we figure out how much a win is worth to a team. Next, we figure out how much a given player contributes to his team's wins. Finally, we convert the number of wins the player is worth to a dollar value. This seems straightforward, but (as we will see) the actual mechanisms behind each step are up for debate. In part I, we'll go over the two main methods for calculating the dollar value of a win. Next, in Monday's entry, we'll use these calculations to determine how many wins Napoli and Victorino need to produce to make their contracts worthwhile. We'll compare these numbers, just for the hell of it, to the same numbers for recently departed players like Adrian Gonzalez, James Loney, and Mauro Gomez, all of whom the Red Sox at some point had considered sticking at first base in 2013.

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The first step is to determine the value of a win, and already we have two opinions on the right way to do this. The first comes to us from FanGraphs and is comparatively simple: a fixed dollar amount per win above replacement. In this model, a 6-win player (i.e., a player who produced six more wins than his replacement counterpart, and typically an all-star) would be worth three times as much as a 2-win player (a mediocre starter). This model suggests that every win produced in 2013 above the replacement level will be worth approximately $6.5 million.

The most glaring issue with this is the notion that all wins are created equal: a team's 102nd win, the model claims, is worth basically as much in revenue as its 72nd. The alternative is a model advanced by economist and professor J.C. Bradbury in "The Baseball Economist", which estimates the marginal revenue product (MRP) for each player. This is defined as the extra revenue a player brings in through his contributions.

Bradbury computes a regression to determine the relationship between revenue, run differential, and the size of a team's market. Since his book was published in 2005, I ran my own using the revenues from the 2012 Forbes Business in Baseball report, run differentials from the 2011 season, and population from the 2010 U.S. Census*.

** - Market sizes were estimated as the population of each city's Metropolitan Statistical Area. Of course, I used the Canadian census data to find the population of Toronto. While we're at it, all values are in USD.*

I came up with the following updated formula:

Revenue in millions = (0.0018 x Run Diff^2) + (0.183 x Run Diff) + (5.698 x market size in millions) + 160.3

(The adjusted R^2 is 0.67, and all coefficients have p < 0.05.) Now we have a formula to estimate a team's revenue based on its performance and market size. We can also estimate the value of each run: a team with a run differential of +1, for instance, would expect to earn approximately $185,000 more than a team with a run differential of 0 in the same market. We can also estimate the revenue for an average (i.e., 81-81) team in the Boston market at $186.3 million*. By measuring each player's runs produced (or saved) above average, we can then determine how much revenue he generates, and use this as a measure of his value above the average player.

** - This number, admittedly, seems awfully low considering the $310 million revenue Forbes estimated in 2012. As it turns out, the Red Sox are something of an outlier and will consistently finish above that curve thanks in large part to their lucrative television deal and their slavishly devoted fan base. This will probably help later, and give a little extra leeway to any borderline contracts.*

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