Prerequisites for derivation: Data.
What Is Replacement Level?
Earlier, we noted that there's an inherent problem in using league average as a baseline: playing time isn't properly weighted. This means that using league average based statistics by themselves don't give a good idea of actual production on the field; they need to be combined with some measure of durability in order to use them as intended. Clearly, this isn't the best arrangement.
How do we deal with this? The MLB minimum salary gives us an important clue. $400,000 represents the zero level of marginal salary a team can commit to a player. Since they're obliged to pay that money to someone anyway, it doesn't really matter to whom it's actually going. Giving a player a job for the minimum means you've acquired him for as close to free as you're ever going to get in baseball, especially as you haven't expended other resources (other players, generally) to get him. Now, if a player's willing to sign for free, that's probably a clue that he's not very good. In fact, he's so bad that other teams don't care if he gets picked up, because there are so many players of that calibre that acquiring one of them has virtually no effect on the size of the talent pool. If another team did care, there'd be some sort of competition to acquire his services, and as a result the team winning control over said player will have had to expend marginal resources to do it.
The above gives us a neat little definition for the league's worst players. We can define a replacement level player as one who costs no marginal resources to acquire. This is the type of player who would fill in for the starter in case of injuries, slumps, alien abductions, etc. If we use replacement level as a baseline for our 'runs above <x>' statistics, we introduce durability into our statistic, and we're therefore measuring marginal productivity, which is what we want.
How Good Is a Replacement Level Player?
We know what we want our baseline to be, at least in philosophical terms. But we don't know how it relates to league average yet. How do we get from the idea of a replacement level player to the idea of what to expect from the average replacement level player? The easiest way is by going back and looking at how well players who've been acquired for 'free' have done, relative to the league. When you do this (for batters - pitchers are an entirely different story) you end up finding that replacement level players are around twenty runs worse than league average per 600 plate appearances (or 0.03 runs below average per plate appearance, if you like rate stats). We now have our baseline set, but there are still some questions left to address.
First of all, there's obviously no way to guarantee that any given 'replacement level' player will actually perform at replacement level. This does not reflect a fundamental problem with the concept, but it's something to bear in mind. Being a replacement level player doesn't necessarily mean replacement level production. Claims that replacement level players are not actually freely available have some validity, but only very specific circumstances (and even if it applied globally, that would simply mean that our definition of 'replacement level' is too high).
Secondly, there are circumstances in which a backup replacing a starter might be significantly better than replacement level. This does not mean that the starter is inherently worth less than he would be were the backup not in place, because replacement level applies across the entire league rather than to one team - trade value should not be affected by the quality of the backups. Thirdly (and related to point two) is the idea that a replacement level player won't get starts and a bench player will. While this is true in some cases, it's hardly a rule, and by using 'bench player' as our benchmark we don't have an economic leg to stand on - we know how that replacement level means free, but 'bench player'? Not a clue. So after converting to bench, we'd have to convert to replacement level anyway to determine how to value these players.
Things to Note
- Again, the -20 runs per 600 PA only really applies to batters. Pitcher replacement level is better measured in expected wining percentage (the values end up being .380 for starters and .470 for relievers, who are extremely easy to replace)
- A replacement level team would cost $10M and win about 48 games (give or take some to allow for luck).
Runs above replacement; wins above replacement.