As shown by the source given in my answer to this question, power consumption is based on the amount of rooms. However, it doesn't go into detail... Now, thinking about possible layout strategies, I want to know:
Do elevators use power?
By "use", I primarily mean if they increase consumption of power. I recently restarted, so I am in no position to test this (it could be tested by looking at how fast power depletes before/after building a bunch of elevators).
Has anybody tested this?
I have a vault with only 4 layers and built a column of elevators down to the deepest level. I did not notice power usage increase of any significant amount.
I conclude from anecdotal experience that elevators effectively do not use any power.
Elevators do not cost any power. You can see the actual numbers of your power/water/food consumption by tapping on them. See the below images for before and after adding 2 elevators to the beginning of a new vault.
As you can see, the vault still had the same power requirements/maximum, and the slider bar of minimum power required didn't move. It's the slider bar that shows the requirement (ie below that and you start to lose power), so if the slider bar doesn't move when you add an elevator, then it doesn't require any additional power.
Edit: I should point out that I'm aware that the numbers themselves do not indicate this -- they are merely a record of current and available power and speak nothing about how quickly it's used.
You can actually see how much food/power/water you're using and producing in the menu, above the gears icon is a little book icon with all the stats for the game. It's very useful. It will tell you how much power you're consuming and if you look you'll see that adding elevators doesn't increase power consumption
