There are better foods out there for your money than Purina. If you've got a local Costco the Costco brand is usually about $25-27 for 40 lbs.
It's made by Diamond, and a lot of feed stores carry the Diamond Naturals line which is identical, the same formula and everything. Around here it's even the same price. It doesn't contain wheat, soy, or corn which are all common food allergens and your dog can't digest them anyway, along with having a higher meat content. You actually end up feeding a little less because the dog doesn't need to eat as much to get the protein and calories it can use.
Another thing to consider is household cleaners. We don't use chemical type stuff in our house so have never had an issue with it. Mostly we mop with stuff like baking soda and use that Dr. Bronner's soap on everything. It's super concentrated and makes the house smell minty fresh.
Is his hair falling out all over? Or is it concentrated in certain spots?
Another thing could be yeast infection in his skin. We fostered a dog with a systemic yeast infection. He smelled sort of, well, yeasty. Like bread almost. His hair was falling out and he was covered in hot spots, along with having really sore joints and icky ears. That was pretty well eliminated by putting him on a grain free diet (Natural Balance Duck and Potato).