When your customer is dissatisfied, there are only two possible outcomes. Either the issue is resolved to the customer's satisfaction or it isn't. Trying to reduce or outsource your customer support costs might actually cost you a lot more in the long run, as you lose customers and they badmouth you to all their friends.
Consider what happens when you focus on costs in your customer service operations:
The cheapest customer support call is the one that doesn't happen at all. Of course, you don't know why the customer didn't call you. Maybe they were happy with your service, or maybe they just didn't have the time or energy to fight their way upstream through your draconian support system. There's just no way to know.
If you're lucky and they do call you, there are still two possible outcomes. You satisfy the customer or you don’t. Probably the cheaper outcome from a functional, departmental, cost-accounting perspective is the one where you quickly tell them it's not possible or it's not your department. You can mark that "resolved, not my department" and forget about it. Maybe the customer will give up at that point.
If you're still lucky, your customer will be patient as you bounce them around your system till they find the department that might be able to solve their problem. Once again you have two possible outcomes. You satisfy the customer or you don’t. Of course while the customer bounces around they might get a little frustrated, but that's okay, because that's not your cost, it's the customer's cost. It's an "externality" and it doesn't show up in your cost-accounting system.
Now that the customer has finally reached the right department, you still have two possible outcomes: you can resolve their problem or not. It's probably going to be cheaper not to solve their problem, because most of the time solutions have some kind of cost. You might have to accept a return or credit the customer's account. If the situation is not in the service rep's rule book, the service rep might not be able to help.
If you're lucky, the customer will ask to speak to a supervisor. At that point there are still two possible outcomes. You can solve the customer's problem or say no. It's probably cheaper to say no, for the reasons outlined above. But maybe the supervisor will say yes, in which case there's a chance your customer will be satisfied.
Customers will put up with this kind of treatment only for so long. Eventually they will find another company that treats them better. That would be great for you though, right? Because the cost of serving someone else's customer is zero! Yay for the cost savings team!
Then, when your customer tells all their friends about their experience, you will lose more customers. Your call center costs will continue to go down. At some point when the last customer has left, you can eliminate your call center altogether. Total cost victory achieved.