Atomic does a great job with transparency around time and money. Their project management style is very communicative so we always knew exactly where we stood in terms of what had been done and what our remaining budget was.
Andrew Burroughs, Healthy Minds Innovations
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Time & Materials
Why budget at all? Just have your team start working, and see where you end up. (Not recommended.)
Carefully adjust the scope, in either breadth or depth, incorporating new knowledge and ideas.
You can ignore the surprises and learning, since you're locked into your original, detailed plan. Or you can renegotiate your job with a series of change orders.