If they have their American Kennel Club official certificate papers, yes — they can run up to $600.00. The papers (which the breeder provides) mean that the parents of your Chi are registered with the AKC and that the puppy is registerable. Additionally, some breeders provide pedigree papers which trace the family history beyond the immediate parents. These papers are vital if you plan to enter your new dog into exhibitions and competitions.
These certificates and papers ensure that the dog is a purebred; however, I hasten to add that just because a dog has papers, doesn’t mean it qualifies for exhibitions and competitions. (See AKC Chihuahua Breed Standards.) Additionally, papers do not necessarily insure that the breeding Chis were well cared for or that they are free from genetic defect.
Chihuahuas who do not have their papers are not significantly different than those with papers. Moreover, they can be bought for much less.