This is a common claim made by teams, but I'm skeptical.
According to economic theory, the value of any given player/coach/trainer on a profit-maximizing team is what that person contributes at the margin to the team and what fans are willing to pay for that marginal contribution. If, for example, Adrian Peterson contributes 1.5 wins to the Vikings and Viking fans are willing to pay $3,000,000 for each extra win, then AP is worth $4,500,000 to the Vikings.
In a profit-maximizing world, the only reason a team would use "money generated by the stadium" on the team is if the stadium enhanced the team's talent or if it increased fans' willingness to pay for the product on the field.
I think that it's likely that a new stadium would enhance fans' willingness to pay overall, but it's not clear that they are willing to pay more to see the action on the field per-se. They may be willing to pay more to enter a new stadium to experience the "newness" of the facility, to sample new concession items, or to experience something else that has little or nothing to do with the action on the field (like a huge scoreboard (like in Arlington, Tx), a giant pirate ship (as in Tampa), or nice views of the surrounding city/geography (as in so many facilities around the country)).
Consider a simplified numerical example. Suppose that the average fan, in an old stadium, is willing to pay $50 to watch the game and an additional $25 for stadium amenities. This fan would be willing to pay as much as $75 for a ticket to the game.
Now suppose that the same fan is willing to pay $50 to watch a game per-se in a new stadium and an additional $50 to experience the new stadium's amenities. This person would be willing to pay as much as $100 for a ticket to a game, a $25 increase. But his/her willingness to pay for the action on the field is unchanged, so the team has no incentive to spend the additional $25 it receives on "competitiveness."
The interesting question from my perspective is "are stadiums and talent complementary?" I think stadiums can be designed to impact the action on the field to improve the chance the home team will win. Here's a blog post I wrote over at The Sports Economist a year and a half ago on this subject. In that post I cite one paper in a professional economic journal that has looked at this question (this one). The paper finds little evidence that a new stadium improves the performance of the home team on average.
That's my response (edited for typos that I found later and with links embedded in the text) to a question posed to me by a reporter writing a story on the Vikings' quest for a new stadium. What I was referring to in the first sentence above was the common claim that because of the small size of the Vikes' market, money generated by a new stadium has to go towards investing in the team.
I touched on the question of whether stadiums and talent are complementary, but there is one other facet that I feel I should point out, and that's the question of do owners of professional teams (or AD's at colleges) maximize profits. I think that the answer is a resounding "yes" in American sports, but I do not doubt that some owners have run their teams as philanthropic organizations to some extent (Ewing Kaufmann, late owner of the Kansas City Royals, comes to mind) and some owners simply enjoy owning teams, much like I enjoy watching a college football game.
Cross-posted at Market Power