Why not create a public bank?

Why don't we have a public bank already? I mean a bank run by the federal government that competes with private banks, keeping them honest.

Let's recap what's happened in the past few years. First, the financial world collapsed (the risky sub-prime loans, CDOs, etc.). So, the federal gov't was forced to loan insane amounts of taxpayer's (and our children's future) money to the banks, just to barely keep them afloat. By and large, that worked: the banks have survived and recovered.

Yet, today they still pay out insane bonuses to their top employees, still fly around in private jets, throw lavish parties, travel to exotic places, etc. They are not extending the loans to small businesses that are required for our economy to really recover. And they are now spending lots of money, fighting the legislation that would regulate things to prevent a future collapse from happening again. In short, they have not changed.

Something has gone terribly wrong!

See, these banks don't contribute directly to economic progress. They don't build houses, invent new products, grow food, create more fuel efficient cars, etc. Really they are just the "lubrication" to enable all the real progress in our economy. They are not suposed to make tons of money, yet they do. And they most certainly shouldn't be given the power to hold our economy hostage, as they are today.

So why not create a public bank, run by the federal government, that would extend legitimate loans with reasonable terms, today? This would put a strong competitive pressure on the existing banks to lend, as the public bank would otherwise take customers away. It would be a much more direct way to stimulate the economy, than the "give lots of money to the banks and hope they loan it out instead of paying themselves fat bonuses" approach that is clearly not working very well today.

It could be a temporary creation, only around until the private banks start behaving well again. Or it could remain indefinitely, keeping the banks honest over time.