How much is one great player worth to your team?
Q. How much is a superstar worth?
A. Short answer: a lot less than you think.
Unless you are a baseball analyst by profession, I would guess that you don't have an instinctive feel for the answer to this question. Let's cheat on the answer a bit by asking for a clarification ...
A. ... compared to what?
Q1. Well, how valuable is a superstar like ARod, Bonds, or Pujols compared to an average major league player?
Q2. How valuable is one of those same guys compared to a replacementlevel player, a guy barely holding on to his job?
A1. Short answer (1) about three or four games.
A2. Short answer (2) about seven games.
Q. What? The Rangers' owner spent $25 million per year to win seven games?
A. Yup, that's just about the size of it. Maybe eight instead of seven. Compared to the guy he could have brought up from TripleA, the Rangers owner paid about three or four million per win to get ARod.
A. Now for the long answers.
Proof 1: the sum of all peers.
Actually the six or seven game figure makes a lot of sense if you think about it. After all, how much could it be?
How many players on a team? Assuming full time duty on the old preexpansion schedule (154 games with no DH), a baseball team consists of 14 players: 8 fielders and 6 pitchers. The fielder portion is obvious. The pitcher portion is more of an imputed statistic. Nine innings times 154 games equals 1386 innings, or six guys pitching 231 innings each. That's sixfull time jobs, even though it is usually divided among more parttimers.
Since a team has fourteen fulltime jobs for 154 games, the most it could average out to is eleven games per fulltime player  exactly eleven games, by a strange quirk. But eleven games per player is the difference between a team which wins ever game, and a team which loses every game, and those things do not exist. The actual difference between the best winning record of all time and the worst is not 1.000, but only .513 (.763 versus .250). I guess you are probably ahead of me on the algebra. 1.000 is to eleven games as .513 is to 5.6 games in the 154 game schedule (therefore 5.9 out of 162 games).
That range of 5.6 to 5.9 games is a bit too low to answer the original question, since it doesn't represent the theoretical maximum between fourteen stars and fourteen replacement players, but merely the greatest difference which has ever occurred. That best team didn't consist of all super stars, and the worst team didn't consist of all replacement players, so let's look at how much greater the difference could have been. Well, in theory.
If you assembled a team of players as good as ARod, would they win 162 games? No. I suppose they would win about 131 or so. The 1906 Chicago Cubs came very close to having a team full of ARods, and they went 11636, equivalent to 12339 with a modern 162 game schedule.
There were only two of the fourteen fulltime jobs where the Cubs could have improved.
As it turns out, the Cubs almost did have ARod at every position! If those same Cubs had managed to acquire Wagner and outfielder Sherry Magee, we could call them a perfect 70 out of 70 for the fourteen positions, on our scale from 14 to 70. As it is, we can assign them a 66 without too much problem. Maybe you could argue for 64 or 67 or something, but the 66 is in the right ballpark. In terms of our modern 162 game schedule, the Cubs won 123 of those games, so how many more would they have won with Magee instead of the platoon guys and Wagner instead of Tinker. Maybe eight games? Even 13131 looks unattainable to me, but let's grant it for now.
On the other side of the coin, the 1962 Mets came very close to having a whole squad of replacement players, and they went 40120. Of the Mets' fourteen players, or playerequivalents, eleven were clearly at the replacement level, worth one out of five on our scale. They had three players who were better than that level:
That gives the Mets a score of 21 on our scale that ranges from 14 to 70. So how bad could they have been with replacement players instead of their three average guys? Let's guess at about nine games. That would have left them 31131 for a 162 game schedule.
131 wins for the slightly improved Cubs, versus 31 wins for the slightly diminished Mets. The difference between a team of ARods and a team of replacements? 100 games in 162.
100 games divided by slightly more than fourteen fulltime players ("slightly more", because we used a 162 game schedule to get the number 100) is about seven games per player.
(Looking at it another way, the difference between 14, the bottom of our scale, and 70, the top of our scale, is 56. The difference between the '62 Mets and the '06 Cubs is not the full 56 points of the scale, but 45. Earlier we mentioned that the actual difference between those two teams was 5.6  5.9 games per fulltime player. We now are able to estimate that range represents about 80% of the theoretical maximum between having the allstar at every position and having a tripleA replacement player at every position. The true gap, then, would be between seven and seven and a half games.
Proof 2: The Black Sox.
You remember the Eight Men Out? They played in 1920, but not in 1921. They were banned from baseball and could not come back for the 1921 season. Therefore, we can calculate a value per player by asking "How good were they?" and then "How much did the team decline when they were banned and replaced with bush leaguers?"
The sum of the values of the first six men, the six regulars who played, is 25/30. Since an allstar is a five pointer on our scale, we'll estimate that those 25 points accumulated by the six players are the equivalent of five full allstars. (A calculation which is completely fair, in my opinion.)
So the Sox lost five allstars and had to replace them with schmucks from the minors. How did the team do? In 1920 they won 96 games. In 1921, without the five stars, they dropped to 62 wins. In other words, losing the five men cost them 34 games. That's seven games per player.
Proof 3: The "Pythagorean Theorem."
Various researchers, primarily Bill James, have determined that the ratio of a team's wins and losses is equal to the square of the ratio between their runs scored and runs allowed.
Excluding defense, a replacementlevel shortstop, a guy like Rey Ordonez, who typically pulls a .300 OBP and a .333 slugging average, will create about 60 runs per year, playing every game. ARod, in the same circumstances, with his .400 OBP and .600 slugging, will create about 145 runs. (I will not go through the mathematical mechanisms used to arrive at these calculations. Read the works of Pete Palmer or Bill James to understand the theoretics of Runs Produced and/or Runs Created.)
The difference is 85 runs.
Assuming their defense is equal, ARod affects a team in today's context as follows:

Runs Scored  Runs Allowed  Wins 
.500 team with replacement level guy 
800  800  81 
Same team with ARod instead 
885  800  89 
Difference 
8 
By this method, the difference seems to be eight games instead of seven. It just so happens that the difference between ARod and the worsthitting major league shortstops is somewhat greater than the difference between Helton and the worsthitting first basemen. If you repeat the calculation for each of the the eight position players, your average would be below eight, close to seven again.
Seven games. That's all that separates the best guys in the majors from the worst at their positions.
There are practical applications to these calculations. Suppose you run the Boston Red Sox and finish this season ten games behind the Yankees. Perhaps, you see no reason for next year to have any significant changes from the normal course of yeartoyear personnel shifts. You have to make some big moves.
Therefore, you know that you should either make two big personnel moves or none.
If you are ten games behind this year, and can replace Bellhorn and/or Mueller with Soriano at second base, and do nothing else, you have no realistic hope of winning the pennant. You may increase by five games or even seven, but you will spend all that money to finish second again, unless luck smiles on you. You can't close a ten game gap with one move, unless I am your left fielder and you can replace me with Barry Bonds.
If you can also replace your fourth starter with Jason Schmidt  then you're cookin' with gas!
From a financial perspective, this spending has a cumulative positive effect, given a hurdle to clear. Essentially, that hurdle is the postseason.
If you spend enough to go from being the 14th best team in the league to the 5th best, you may or may not see a return on your investment, depending on how your team attendance responds to that improvement.
But if you can spend a little more, getting yourself into the postseason, you get higher attendance per game, and more games. Your revenues increase substantially.
If your team is already assured of making the postseason, your investment is not as valuable as the money spent by a team that goes from noncontenders to contenders.
Therefore, the greatest return on your investment presumably occurs when you spend to go from "out of contention" to "in the postseason", provided, of course, that the strategy works.
The Rangers did not pull into contention with ARod, and it would be very difficult to calculate much, if any, return on that massive investment.
How much is one great pitcher worth to your team?
About the same. Great pitchers don't generally save 85 runs for their teams compared to a replacement level guy, but they don't have to A pitcher's run contribution is more important than a hitter's. He doesn't have to save 85 runs to be as valuable as a hitter who adds 85 runs. He can do accomplish the same thing with 68 runs.
Here's the math.
Point one. A defensive run is more important than an offensive run.
That makes perfect sense of you think it through. It is very obvious in extreme examples.

Runs Scored  Runs Allowed  Wins  Value 
.500 team 
800  800  81  
Same team scoring 800 more runs 
1600  800  130  49 wins 
Same team allowing 800 fewer runs  800  0  162  81 wins 
In realistic examples, the difference is less dramatic, but real. You need to see the decimal places to estimate it accurately.

Runs Scored  Runs Allowed  Wins 
.500 team 
800  800  81 
Same team with 85 additional runs 
885  800  89.2 
Same team allowing 85 fewer runs 
800  715  90.6 
Same team allowing 77 fewer runs 
800  723  89.2 
You can see that, in today's hitting context, 77 pitcher runs are equal to 85 batter runs.
Point two. The same amount of runs has a greater impact if exerted in fewer games.

Runs Scored  Runs Allowed  Wins  Value 
.500 team , 162 games 
800  800  81  
Same team allowing 77 fewer runs 
723  89.2  8.2 wins  
.500 team, 32.4 games  160  160  16.2  
Same team allowing 77 fewer runs  160  83  25.5  9.3 wins 
Same team allowing 68 fewer runs  160  92  24.4  8.2 wins 
A pitcher making 32.4 starts (exactly 1/5 of his team's games) can achieve the same thing (adding 8.2 wins) by preventing 68 runs as an everyday player can achieve by adding 85 runs. It is coincidental, but serendipitous, that 68 is exactly 80% of 85. In today's context, one run contributed by a batter has only 4/5 of the value of a run contributed by a pitcher. If Jason Schmidt can save you 68 runs, he is as valuable as ARod adding 85. In both cases, their contributions will add 8.2 wins to a .500 team which scores and prevents 800 runs per season.
It is no simple task for Schmidt to prevent 68 runs. If he pitches 204 innings, he'll need to have an earned run average three runs better than your replacement guy. It's difficult, but it is possible. If your replaced guy is allowing a 5.34 era, which is entirely reasonable, Schmidt is certainly capable of a 2.34. He did it last year! The Giants fifth starter, Jesse Foppert, actually had a 5.03, so it is not at all foolish to assume that the hypothetical guy who would take Schmidt's innings in case of an injury would be slightly worse than Foppert, maybe ... oh, I don't know. Maybe 5.34
I guess you didn't really need me to do any math with pitchers. Schmidt was 175 last year. A replacement level guy would have been about 913 in the same 22 decisions. You could have figured out Schmidt's eight game value in a minute, and I spent hours on it. Ah, well.
The bottom line is that a great pitcher has about the same ability to help you as a great position player. Assuming they will replace marginal innings and atbats, ARod and Schmidt are probably about even in terms of their ability to improve your record over the guy you have playing there now. Each of them is worth about 8 games to you. An average allstar, at a tier below those two guys, will be more in the six or seven game range.
What about the value of a fielder?
Who cares?
Just joshin'.
For the record, the value of fielding runs is about halfway between that of pitching and batting runs. Remember, runs prevented are more valuable than runs added, so fielding runs receive that benefit, but they do not get the value of being compressed into fewer games, ala pitching runs.
Amount of runs necessary to produce 8.2 wins at 800 runs per team per year.

Runs 
Batter, 162 games 
85 
Fielder, 162 games 
77 
Pitcher, 32.4 games  68 