Will a shorter bar increase my chainsaw's power significantly?
No. Or yes. 

It isn't like gearing. A shorter chain doesn't compound torque like a smaller gear does. The amount of force applied at any point on the chain is identical. 

However, the shorter bar limits how much contact the chain can make with the wood. Herein lies the difference. If you're cutting 16" of wood rather than 18" the motor isn't working as hard. 

Should you do it? Probably not. You needlessly restrict yourself with respect to reach, and you can accomplish the same goal by working in a less aggressive manner when you're dealing with maximum-depth cuts.