If you're in a space hotel at 22,000 miles, and the cord breaks, what happens? You should just hover there until you can be recovered, right? Or would you be thrown out into space?
If that hotel in on the space end of such a elevator, then you go flying off to a higher orbit, because the thing isnt' really in a geostationary orbit at 22,000 miles, because the stress in the cable is providing an additional force on the "hotel", and so it will orbit be orbiting faster than a regular circular orbit would at that altitude.
One way to envision the situation would be to imagine an object connected by two cords to fishing reels, each with a different drag setting. If one cord broke (the other representing the gravity force, which doesn't "break") the other cord would be pulled out until the force on it was the amount of the drag setting. Even though centrifugal force doesn't really exist, the situation can be most easily explained by invoking it. A longer string, that is a higher orbit, means a lower centrifugal force for the same speed. F = m (V^2/R). where V = speed, m = mass, R = lenth of the string or orbital height.