BW,
If you have that kind of control over clients, consider yourself blessed. Since my days as an ad agency art director buying all kinds of production we ALWAYS had to work with a budget, so any production companies that bid on our work knew the amount going in. This of course eliminated many big players, who simply could not be touched for our $s. In my example above, I was bidding for an ad agency, where in many cases today it is still “this is what we’ve got, you’re either in or out.” It’s probably worse now than ever, because of all the competition caused by producers like you and me.
Citing my example further, and some of the other good advice here, it is sometimes advantageous to take a discounted job as a gesture of faith to get more and better jobs from the same agency.
That was my situation, it was an agency I want more work from, and it turned out well for me anyway as they increased the budget mid-job.
Thanks to all of you for some valuable insights, I have to do billing today, I intend to use a lot of your suggestions.