My boyfriend tells the story of a patient he had when he was just starting his career. She complained about the price of the service time and again, saying she could hardly afford it, and begged him to make a special deal for her.
He finally did, only to discover later on that she was one of the wealthiest women in that little Costa Rican town. He felt terrible, because everyone else was paying full price without complaining.
While he does volunteer his services to those in need and occasionally arranges trades with people who can't afford to pay, he's never consented to a discount like that again.
Over the course of your independent career, you will likely run into someone who asks for a lower price. (There's nothing inherently wrong with this: you never know unless you ask, and learning to ask for what you want is a valuable skill.) As the business owner, you will be the one charged with deciding what to do about it.
Remember: what you do has value.
It's fair to expect something in exchange for what you have to offer. If your prices are fair and your work is good, you should never have a reason to feel guilty about accepting money for what you do.
Why is the person asking?
Is she dissatisfied with part of the service? Is he in genuine financial trouble and in need — not merely want — of your service? Is she asking for a trial period, after which time she will be happy to pay in full if she decides to stay with you? Has he been pestering you for a deal, all the while continuing to purchase your service at normal price?
How do you feel about it?
If the idea of lowering your price for someone makes you uncomfortable or, worse, resentful, don't do it.
You don't owe anyone an explanation.
You set your prices after carefully considering the market and the quality of your work. A person asking for a price cut can be directed to any offers you might have ongoing: discounts for buying in bulk, a deal of the month, or a freebie for referring a friend. Beyond that, a polite, "No" is the way to go. If the client can find a better deal elsewhere, you might lose him, but that keeps you available to take care of the clients who truly value your services and are happy to pay the prices you've set.
Have you ever lowered your prices at the request of a customer? How did it turn out?