I prefer not to specify a budget ahead of time, so as not to prejudice your estimate.
Instead, for some reason I took the bait and made a low ball offer. In a post yesterday on her Atlantic blog ("The Naive Negotiator"), Megan McCardle explained the problem with low ball offers:
There is a zone of possible agreement (known to those who study this sort of thing as the ZOPA). You can't negotiate your way out of that zone no matter where you start. Nor does starting from a more aggressive bargaining point always mean that you will do better in the negotiation. It can often mean you do worse, because you poison the process.
My mother used to sell real estate, and you'd see this a lot with stupid buyers, particularly men using newbie agents: they'd submit an unrealistically low bid on the notion that this would force the buyer to bargain down. What it actually did was convince the buyer that it was a waste of time to negotiate with you, and/or make them angry.