What's a reasonable amount of time to expect a client to respond to an applicant before you retract an application for unresponsiveness?
Seems like the question has the context of an online job board, but a similar concept applies to more traditional interaction (such as sending a proposal via email):
Does it matter? It's worth considering whether it matters or not. Actually withdrawing an application seems worthwhile only when it affects some kind of online metric. Perhaps a potentially discouraging bid to success ratio that's public, or (in the case of oDesk) is applied against the weekly limit of bids you can place (although, in that specific context, I'm not sure withdrawing gives you back a bid).
It also signals to a client that you're not interested anymore - which may be a good thing. If it's a potential client that asks a lot of questions, then goes silent, then is back with questions - it may be good to signal to them that there's no more interest (in traditional communication this would be a delicate email, for online job boards, it's a little less personal).
Or, if your concerned about your time available, you may not want to pursue other work while still negotiating a potential contract.
Of course you it may not be a good thing to signal, if the client has just been distracted by pressing needs, and is not just stringing you along.
So how long? Like a lot of things, it's kind of a case by case basis. Does the client just seem unreliable when it comes to communication? Then probably not someone you want to work with. Have they been very active, then fall silent for a few days? Might just be that something came up, and they need a gentle reminder.
How far along are the negotiations? If it's just the start, and there's no other advantages to close communication - I'd leave it open. At that point no party is really expecting anything, and if they come back after a week or so and your availability is gone - they really shouldn't be surprised.
If you're at that point where they've asked you to hold time open, then disappeared before taking whatever action needed to start work, then I'd only give it day or so (business days) until notifying them that I couldn't guarantee the time any longer.
Of course, I'd also consider what my current work load was, and if any other clients were wanting that time.
[A bit of a rambling answer, I know. Will try to tighten it up when I get a chance.]
My first question is "Why would you ever want to retract"?
I'd say 90% of time, you're being offended because the client is not responding. Well, don't be. There are multiple occasions where I was contracted 6 months after my application. Some clients are lazy and it does not mean that the project you're applying to is their priority project.
And once the client contracts you if your price or costs of material have changed, simply inform the client offering him a new price. I faced the same thing with my hours roof repair just because I was lazy - once I called the contractor (3 months later) he told me that the costs are 5% higher because of the material.
Doing business successfully means that many times you'll have to put your anger or disappointment aside.