semi-wild farm/barn cats sure help with the mice. cheaper to feed the cats than repair mouse damage. no rats here in Alberta but the occasional pack-rat around here. they can do damage but they don't breed like the norway rats and they are small enough that a cat can handle them. red squirrels around here can also do a lot of damage but the cats seem to keep them in check
I second the motion for barn cats. Feed them just enough so they stay, but hungry enough to still hunt. Surprisingly, my kids pet cat is well fed, but is also an excellent mouser. She goes out daily and is always seen stalking pray in the garden, under the porch or deck, or in my barn. I have found many of her kills laid out where I can find them, but she never eats them, just lays them in front of our door.
As for the damage, rats are monsters that chew on everything electrical. I left a car in Midland Texas for 2 weeks while I went home. When I got back, it was 109degrees and I suddenly had no A.C., a check engine light and my electric cooling fan wouldn't turn off. That bugger chewed through a bunch of wires under the hood and was building a nest in the vee of the 3.0l v6 engine, under the intake manifold.