I've never asked for a "discount" from a contractor, on other things I have, normally a contractor tries to give you his lowest price, with a little wiggle room. I didn't even know contractors "expected" it, car dealers yes, contractors no. However, I wouldn't be afraid to ask a contractor for a better price. You can be honest and say you have been given a lower bid, and were wondering if he could match it? He'll either say "yes" "no" or give you reasons, why his bid is higher. This can be really important information, such as the actual material he's buying is a higher quality, and why in this particular case that might be important. He might also point out things that are including in his bid, that aren't included in the other bid, all touch up work, painting, etc. By communicating with him in this way, you will learn more about him, and his work ethic. After doing this, I found what sounded like the highest bid, turned out to be the lowest bid, when I again consulted the previous low bidder because certain things weren't included. Be sure to get a completion date in writing, with monetary fines included for each day work goes unfinished over due date, I think no lower than 10% for each day, even at that rate they could leave you hanging for 100 days.
Many contractors start the job and leave you hanging for months while they're bidding on new jobs.