I mean, yes. I'm evaluating the teaching industry as a whole as being more competitive than the sports industry. I think you're mistaking different measures of competition, though, and we're possibly talking about the same thing in exactly opposite terms.

To get us back on the same page: Entering a new field is best thought of as a graph of how much you've been paid overall (the integral of your wages) minus job-related costs (student loans etc.) over time. Probably this time-axis is not "real time" but "time invested," as you may well be working for tips while trying to "break into" either teaching or acting. Now, there is some graph overlaid, which represents the amount you could have made doing other stuff (again in integral form, total profit you could have expected to make by that time pursuing other avenues). This is called an "opportunity cost" and often subtracted off of the original graph in some circles but I prefer to plot them side-by-side. In any case, find the t-position where these two graphs first cross and that (time, profit-to-date) break-even point is the "barrier to entry" into a new field. By the mean value theorem, sometime before you hit that point, the slope of your new career will have the slope of your old one, which is probably the definitive time when you "became a teacher" or "became an actor," rather than simply aspiring and doing stuff-on-the-side to break into the field. And sharper concavity on the curves suggests that this is closer or further from the break-even point.

OK, now that we've got the same picture in our heads: you're either measuring "cutthroatness" as either the time to become a teacher or that total break-even profit, and then you come to a natural conclusion that teaching, being a more gradual path to the point and having a lower point, is less cut-throat of a profession. I totally get that, and it's pretty persuasive.

However, what's salient for the economic theory is competitiveness for people who are already in the relevant industry. And that is where we cross terms, either inexactly (if you meant the first one) or exactly the reverse (if you meant the second one).

In standard macroeconomic theory, the magnitude of that break-even (profit, time) is the major factor determining how competitive the field is for the people who have already made it: a huge break-even investment means that the actors who have made it are going to make better wages.

One easy way to see this is to imagine a profession with no barriers to entry at all, something which anyone can just decide to go and do. (The stereotype of "fry cook" comes to mind.) In the long run, those are necessarily the least profitable professions, because anyone in a less-profitable profession will effortlessly switch to those professions, which will dilute profit in those professions and concentrate profit in the professions they're leaving, tending to equilibrate the two. Barriers to entry, whether those of copyright and patents or that of needing to fight for years to be recognized for your acting prowess, are the only things stopping the unwashed masses from coming in and making sport of it; it stands to reason that the bigger the barriers, the more of them are stopped.

In that sense, teaching is much more cutthroat than acting or sports or being a doctor; once you've made it as an actor/actress or an athlete or a doctor, you're now in the industry and therefore you're getting paid well, even if it cost you a lot to get in. You can enjoy a healthy security now that you've made it, precisely because it was so hard to make it, and everyone else needs to climb that mountain, too.

But if you're a teacher, your chance of having your school close or getting fired -- erm, "not asked to return" -- the next year is much higher, and you (a) know it and (b) often don't have the safety margin to easily afford it. You are much more replaceable than a professional athlete, therefore you have much less negotiating power at the table.