What is the purpose of including dogs in the story?

When the tinker arrives at her farm, his mongrel dog comes first, running ahead of the wagon. After observing this, Elisa's two dogs immediately run forward, threatening the dog, who eventually cowers back under the wagon, unharmed but nervous.

The interaction between Elisa's dogs and the tinker's dog is symbolic of the interaction between Elisa and the tinker themselves. Just as her dogs are stronger than the tinker's mongrel, so is Elisa wittier, smarter, and more of a robust person than the tinker. She demonstrates superior wit during their banter, and, as she later reveals, she is just as capable as him of doing any of his repair work. However, despite her superior wit and skill, Elisa still succumbs to the tinker's charm, paying him for a job she could have done herself, and he leaves, just like his dog, unharmed and intact -- and fifty cents richer.

As the tinker's wagon rolls away, Elisa's dogs have abandoned the threat of the mongrel, and are sleeping. In the same way, Elisa has passively allowed the tinker to extort her out of fifty cents, and leave with her money in his pocket and her flowers in his wagon.