I quoted for a job 3years ago to lay oak flooring, the customer had already bought it & if I remember correctly had paid around £3000 for it. I quoted £3200 to fit it. I received a phone call one day saying he had another quote for £1800 but he would like me to fit it if I could lower my price a little.
He went about it in a nice way & I too wanted the job but said,
"No, that's my price, the other guys price seems awfully cheap & if it's not done right he could ruin your flooring that you've spent £3000 on & you will have wasted a lot of money if it's a mess, have a think about it & you can get back to me, get another quote if you like & compare that"
So I left it at that, he called me back 10mins later to say go ahead, I presume he just had a chat with his wife about it. He said later he was glad he went with me as the other guy did not really inspire a lot of confidence & I appeared to know what I was talking about.
The way I look at it is if they are asking you to come closer to someone elses price, they must want you to do the job, otherwise why did they just not go for the lowest quote in the first place, it's all about selling yourself. I have no idea if the other guy was any better or worse than me.
I have since done other work for the guy, so he must have been happy. You hear so many horror stories out there about poor tradesmen, so assure them you are good at what you do & know your stuff. I just try & do a good job & provide a good service while making a good living at it.