When my wife and I bought our home, we had to have a living room, a dining room, a family room, 4 bedrooms, a 2nd floor hallway, and a stairway re-carpeted. Jumping ahead, I can't guess what it would cost to install carpet on stairs, but ... We removed the old carpeting ourselves and saved some $$$, tack-tracks and all.
For our stairs, the stairs were installed with the carpet being layed over the tread and tucked underneath, then run down to the next riser, and tacked there too. Like in Bruce M's description, it's the opposite of the over-the-tread-down-to-the-bottom-of-the-riser install. But in our case, the carpet was standard carpet (not a 'thin' type of carpet), and our carpeting is expected to last 8-12 years depending on your level of carpet-care.
As I mentioned, I can't tell you what it would cost to install carpet on stairs with either method described here or elsewhere, or with either type of carpet. But, also like I mentioned, our carpet was installed in the tucked-under-the-tread method as opposed to the not-tucked-under-the-tread method. And, I must say, it looks nice because you can still see the profile of the stairs! This is aestetically appealing though I can't comment otherwise. One thing we did was tip the installers (with $$$ and sodas) when they came to the house to do the 4 bedrooms, the upstairs hallway, and the stairs. To my surprise, I found out that the foreman on the 3-man crew was 60 years old and NEVER was tipped BEFORE he did a job! He said he'd been installing carpet for 35 years and was never tipped before a job. Well, the retailer that sold me the carpet called me up the next day and said, "What did you do to my installer?! The man NEVER talks and today he wouldn't shut up! He said you tipped him BEFORE he did the install and he was so excited!" Certainly, I was happy to make the day of the 3-man crew. Doing the install with this method left me with a truely professional-looking job. I mean, you should see the detail. I guess what I'm saying is the same as Mike and Bruce. It's not as easy as you think. And a little $$$ goes a long way. By tipping before the job begins gives the workers a bit more incentive to do a good job then having to wait until they're done where they may NOT get a tip. Try it; you may like it. And I tip like this pretty much everywhere, except as you guessed it, at the restaurant. I gave each of the workers $15 on a $1400 job. $45 is a cheap price to pay UP FRONT for a little incentive to do more than a routine job.