Some live wallpapers are more battery intensive than others, as well. It all depends on the amount of animation, whether it runs in the background while the screen is off, and how much processing it has to do. The least battery intensive are those that are passive or static unless/until you interact with them. I wouldn't worry about it too much, though. Unless you notice abnormal battery drain after installing one, you likely won't be able to tell much of a difference.
I take for granted that live wallpaper on your Amoled screen is like having a video running constantly in the background. Especially if the live wallpaper has bright colors. I haven't really experimented much with live wallpapers and battery drain. Just putting 2+2 together. Personally, it's very annoying to have things move around while I'm trying to find an app.
Nothing like a regular video running. As to things moving around while you search for an app, the live wallpaper only runs on your home screen. It does not run in the app drawer.
Tell us.......What is the difference between a live wallpaper and a video running in the background? I'd like to know.
If you keep apps on your home screen, you do see your live wallpaper. And on many phones you can have live wallpaper in the app drawer.
Unrelated but I haven't used my nexus 7 in awhile and found it dead which is said to be bad for the battery. Should I be worried because I think a battery cycle or 2 has been used which diminishes it's capacity?