I wonder if this is something that is commonly associated with this breed of dog. The reason I say this is our first Chihuahua Buddy had the same problem. He actually had dry red patches of skin that he would itch and it caused him to start losing his fur as a result.
My brother took him to the vet and Buddy was given a special lotion to use. Only problem there were times when he wanted to lick it off and we weren't sure if it would be harmful for him. My brother started to use Aloe Vera gel,in its purest form. What I mean by purest form is some of the ones commonly seen in stores contain alcohol. So, we went online and found one that was 100% natural. This was important because if Buddy tried to lick it off, the Aloe wouldn't be harmful to him according to the vet. In the end the Also Vera did help and he got over the skin irritation. We were never sure what really caused it, and he never got it again.