You're kidding right? It's the best thing to do. Economically and morally. Same goes for liberals. :)
How much does hunting down, capturing, and neutering cost? I'll grant that there are even more ridiculous ways to waste money (even confining the subject to the population control of animals), but it's positively absurd... just like capturing and sterilizing deer.
And morally? Only if you've decided that it is so and incorporated it into your moral code - most commonly by people that confuse "Bambi" with a real deer. Most animals place in life is to be torn apart and eaten. That we choose to amuse ourselves with a relationship with some of them is an exception, and our care on this account should be limited to those we have an actual relationship with. The "immoral" part of this is the support and over-regulation of harvesting of an invasive species, and the wasting of perfectly good protein. ...not that I have any good recipes.
If I have a pet, others shouldn't harm it, for in doing so, they harm me. For people to revel specifically in the pain and suffering of others, including animals, is problematic. ...but to freak out because every life-form in the universe isn't going to live a painless life and death is itself problematic. ...And this whole argument mirroring the "how can we bring a child into a world with war and poverty, and Nixon?" thing is just sophistry.
To quote myself: "Sophistry...it sounds like thinking." --lepton