I'm not sure if I understand your question.
Are you talking about the wastegate actuator? If so, then yes, it should move, but only when the boost gets to a level where it can overcome the spring pressure in the actuator. When the boost gets high enough (around 12-14 psi for a stock motor) the boost pressure will push against the diaphragm in the actuator and the arm will move....this opens the wastegate puck allowing exhaust gas to by-pass the turbo exhaust wheel and the boost will be in control.
If the actuator...or the arm/puck are stuck you may overboost your engine. Like wise if the puck is not sealing or the arm is too loose the engine won't build boost very well because it is leaking.
The arm shouldn't move on it's own when the engine is at idle. You should be able to move it with your (gloved) hand though. It will feel pretty stiff.
The best thing to do is to take off the c-clip....remove the actuator from the arm. Wiggle the puck arm to make sure it's not stuck...it should move freely. If you have a mity-vac tool you can use it to pump up the actuator diaphragm to see what pressure it actually opens at. Plus it allows you to check for leaks in the actuator diaphragm.
Good Luck,
John