Would a long -haired chihuahua be good for a desert environment?

My parents and I are thinking of getting a chihuahua but I'm debating over a short-haired one or a long-haired one, and like I said before, we live in a desert-like environment.

Chihuahuas are heat seekers. I have both the long and short haired variety. I also live in what could be considered desert conditions. Of course, chis are indoor dogs who don't tolerate cold well at all, but with plenty of water and someplace cool to lay when they've had enough heat (mine like laying next to the toilet on the cool tile in the bathroom!) they're ideal for someplace that would be a little too warm for other dogs.
Incidentally, my 2 short hairs shed worse than my long haired!
They are sweet, happy little dogs. I think you'll be pleased with your choice!