The problem with budgets prepared by law firms for a specific matter is that the law firm will quite commonly (and naturally) budget high. The firm gains nothing if it submits a lean budget and later has to explain why costs rose above the budget.
Theoretically, in-house counsel can pry into a somewhat padded budget with a gimlet eye and reduce it, but that causes ill feelings and no inside lawyer looks forward to that operation.
Here’s the modest proposal – three actually. Have two firms, each capable of doing good work, submit budgets for new matters and go with the lowest. Over time, assuming no collusion by the firms, you can judge results and have some assurance that the selected budget was the more appropriate.
Alternatively, let one of the firms submit a budget and give the other one a few days to decide whether it would work under the terms of that budget or not. This gives a little market insight into the perspicuity of the submitting law firm. If the first budget is generous, the second firm will leap at it; knowing this, the first firm will try to submit a realistic budget figure.
The final suggestion: over the course of a number of matters with a firm that budgets, do 50 percent of the matters complete below budget and 50 percent above budget? If a high proportion come in under budgets, it is likely that the initial budgets were inflated.
These are simply ruminations. I do not know of any law departments that have taken these approaches to budget accuracy.