The linear weights run value of a HR is 1.4 according to Pete Palmer, editor of the Baseball Encyclopedia. But as many writers have pointed out, notably Tangotiger, a HR might have a different value depending upon the context or environment they are in. For example, a HR is probably more valuable on a team with a .375 OBP than with a .300 OBP (on-base percentage) since there is a better chance that a runner will be on base when the HR is hit.
But major league teams have to make decisions about adding (or not adding) players to their team, which means they replace someone already on the team. A team cannot just simply add more HRs. Suppose we take a great hitter, Mickey Mantle from 1961, and we think about adding him to a high scoring team or a low scoring team. The high scoring team probably has a higher OBP and it would seem that a HR added to that team would increase runs more than on the low scoring team, which probably has a low OBP. But if Mantle is added to the low scoring team, he most likely will replace a hitter who is not as good as the hitter he would replace if he were added to a high scoring team. So the team OBP and SLG (slugging percentage) would increase more for the low scoring team and consequently their runs scored per game will increase more than for the high scoring team. After all, the more OBP and SLG increase, the more runs will increase.
To test this, I first came up with a relationship between team runs and team OBP and team power hitting. Instead of using SLG, which is highly correlated with OBP, I used extra bases per PA (plate appearances). The relationship I found came from a linear regression in which team runs per PA was the dependent variable and team OBP and team extra bases per PA were the independent variables. This is a variation of isolated power (SGL - AVG or ISO), which is a better measure of power hitting than SLG, since a guy could get a single every time up and have an SLG of 1.000 with no extra base power. But since I am dividing extra bases by PAs instead of ABs (which is what ISO does), I will call it ADJISO for adjusted ISO. Here is the equation:
R/PA = -.095 + .526*OBP + .327*ADJISO
The r-squared was .902, meaning that about 90% of the variation in team runs per PA is explained by the equation. The standard error was .004. For a full season average of about 6,000 PAs (just walks + ABs), that is about 24 runs a season.
The next step is to have a given player replace the average hitter on a team and see how it changes the team's OBP and SLG (I assumed that the "replacement" player would account for one-ninth of the team's PAs). Then I estimated the team's increase in PAs given that its team OBP has gone up. I used Mickey Mantle from 1961, so he would raise any team's OBP. This follows an approach taken in the 1999 Big Bad Baseball Annual with "extrapolated runs." That was a more sophisticated approach developed by Jim Furtado. But I did not feel like adjusting every stat for a team, like singles, doubles, etc. I think the results will generally be the same, anyway.
Once I know the team's new OBP and ADJISO, I plugged the numbers into the equation to find the team's new R/PA. Once that was found, I multiplied it by how many PAs per game the team was expected to have with the new, higher OBP. I now had a predicted runs per game based on the player that was added. I compared this to how many runs per game the team was expected to score based on the equation before the new player was added (sometimes teams score alot more or less runs than expected due to luck, so I did not use actual runs). Then I found the difference for every team from 1960-1998 (the teams used for the regression). That got multiplied by 162, to get the difference for a full season. As mentioned earlier, the player I added to each team was Mickey Mantle from 1961. The table below shows the top ten and bottom ten in terms of how many runs Mantle would add if he replaced a team's average hitter.
If Mantle got added to the 1963 Astros (Colt 45's), they would score 117.86 more runs. But he would only add 75.75 to the 1996 Indians. This difference is because Mantle would add 18 points to the Astros team OBP but only about 9 to the Indians. The Astros gained about .025 in ADJISO while the Indians gained just .015. Plus the Astros would gain more PAs.
The Astros' gain is about 42 more runs than the Indians. So, although, any given HR on the Indians is probably worth more than one on the Astros, adding Mantle to the Astros will cause a bigger increase in runs. Also, the correlation between the runs a team was predicted to score before Mantle was added (based on the regression) and the increase that the regression predicted was about -.97. Notice from the table that the teams that have the lowest increases were already high scoring teams and the teams that had the biggest increases were low scoring teams.