It's not just that outsourcing is like, as James put it, chasing your own tail. There are other hidden costs involved.
There are practical reasons why an outsourcer's service level is seldom as high as what you'll get from your own people. If outside contractors cut costs, it might be because they're more efficient. But it's far more likely that the savings occur because contractors pay their people less, spend less on training, or both. In the petrochemical and mining industries, for example, research shows that a disproportionate number of accidents involve contractors. Just as there's no such thing as a free lunch, there's no such thing as a free worker who's been properly trained to do a great job.
As I've said before, outsourcing is a short-sighted solution rooted in the current mentality of instant gratification. Outsourcing makes sense in some situations, but cost should not be the sole driving factor. (And no, job security is not one of the reasons ).
Let me remind you that customer service matters. It's really the one on-going interaction your customer will have with you - and it's always in a time of need. Is that really where you want to sacrifice quality?