As a freelancer, you have made this experience more often than you liked to. No matter what you offer and to whom you offer it to. There is ALWAYS someone cheaper. It is like a law of nature. As sure as day follows night each of your offers is doubled by a cheaper one. Keep calm. I know that the other offer is not really cheaper or at least not only cheaper in price. It is more in the way that today’s cartoon depicts:
By the way, we have more cartoons for you here.