Take it one step further and let people specify what their money goes to. Automate the politicians!
It seems like a nice idea on the face of it, but apart from the inevitable chaos which comes from having budgets of government departments fluctuate wildly from year to year, I think it would lead to a world where government departments spend about half their time and money begging taxpayers for money. Every television ad for the first four months of the year would probably be some variation on: "Remember, the FBI do…
Also, I'm not sure why making government departments beg for money and have to live in the free market is on the face of it a bad thing. Would it be any different than the current system, where politicians beg us for money while promising to allocate our taxes a particular way, but then usually fail to follow through completely on? Having the departments beg for money directly seems more efficient somehow.
Of course, I think what would really suffer is bold new infrastructure programs. People would be willing to allocate money to roads once a couple high profile bridges collapse, but would we ever see a space elevator?