As a rule of thumb, when a company is trying to outsource any unit of their operation, it means doing the activity in-house is more expensive. So naturally, outsourcing is cheaper.
While it is a less expensive option, it isn't without its cons. One of the key problems is that you lose control of the quality of service. Someone else is talking and supporting your clients. Someone who doesn't care as much for your company's bottom line. I am of course generalizing and there are a few very respectable and known outsourcing support companies out there who do a great job. So if you are leaning towards this option, at least do your homework and pick a partner with a positive track record. Don't just go with anyone and don't go with the cheapest option, as it might end up costing you a lot more.