Such customers often demand an extremely low price for your products or services. A price which is possibly impossible to sell at. In such scenarios, it's always helpful to do an ROI analysis with the customer. They might spend only 50% of what they're spending now, by going to a competitor, but with that additional cost, you're providing XYZ services / features that no one else can, a dedicated and highly motivated customer support team which thrives on customer satisfaction, the confidence and guarantee of your time tested and proven solution that is COMPLETE in all respects, and the confidence that you'd come to their rescue ANYTIME they need help.

This answer can also be customized based on the specific industry in question.

It's important to remember that you may not always be able to retain a customer in such circumstances. However, never engage in an aggressive standoff, or bad mouth your competition. Hold your ground and trust your product. This customer will come back if you genuinely provide a superior solution.