Only if the reward is more than the cost of breeding the rat.

I grew up in Saskatchewan, Canada. While I missed out, my father always talked about turning in gopher tails for money. My recollection is that my older brother was able to do this, and he got $1 per tail. At that price, even in the 1970s, I doubt you could have made any real money raising gophers.

Presumably there's a sweet spot below the cost of raising a rat but above the cost of harvesting a superabundant wild rat.

Tax the rat farms!