I guess the reason why bitcoin has a stable price is because miners are mining sometimes or even stop mining at all. The supply of coins freeze and the price would definitely freeze to. Correct me if im wrong but i really believe mining and the price are linked. If one stop moving then the other one will follow.
On the contrary, reduction of bitcoin mining should increase the price. The supply decreases and the price increases accordingly. In fact, stopping mining doesn't have much impact on the price. Only human psychology affects the price.