Please. The Fed is in effect a government agency. It was created by government. It's has a secured monopoly status by the government. It operates on behalf of the government (as well as their private member banks and other cronies). It's more correct to call it a hybrid institution. As Rothbard said, one can think of all government institutions as private--in a sense they are, they just have a different business model and organizational structure, and are imposed by the barrel of a gun, working in the guise of "the public interest."
