I understand that the current Federal Bond Buying program ($80 billion a month) is allowing a fixed supply of money to the American Government.
If this is the case, then why did the government "shut down"? Don't they have funds available?
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They shutdown because they didn't have a budget, and the law limits what a US Government agency can spend if they don't have a budget. They can provide certain essential functions: air traffic control, border security, food inspections, military,...; but not other non-essential functions: national parks, department of education, large portions of NASA.
This is not due to a lack of money or access to bonds, it is because the US Congress didn't pass the budget for the agencies.
The other answers address the legal reasons for the shutdown, so I'm going to address this part of the original question and the misconception about the role of the Federal Reserve.
I understand that the current Federal Bond Buying program (80 Billion a month) are allowing a fixed supply of money to the American Government.
This is incorrect. The Federal Reserve's bond buying program, i.e. quantitative easing (QE), isn't related to the US government's budget. Under QE, the Federal Reserve purchases 40 billion of mortgage-backed securities and 40 billion of long-term Treasury securities in the open market each month.
When the Federal Reserve purchases mortgage-backed securities, it purchases them at market prices from private institutions that hold them, like investment banks. When the Federal Reserve purchases Treasury securities, it could purchase them from the US Treasury when they're first auctioned off, in which case the money would go to the US government. In most cases, however, the Federal Reserve purchases Treasury securities on the open market as well, which is why such purchases are called open market operations.
In short, the money spent purchasing bonds as part of QE doesn't go to the federal government. It's injected into the bond market when the Federal Reserve purchases mortgage-backed securities and long-term Treasury securities from other financial institutions.
I want to add a quick note on the structure of the Federal Reserve system, which may help clear up the question too. First and foremost, even though the Federal Reserve has "federal" in its name, it isn't a taxpayer-funded government agency. The Board of Governors in Washington, DC is a government institution, but the regional Federal Reserve banks in New York, Chicago, etc., are not. They're private institutions that are owned by private US banks, regulated/audited through the Board of Governors, the Government Accountability Office, etc., and funded through bank's membership fees, product sales, fees on financial services, and profits on their portfolios.
In turn, the Board of Governors is comprised of political appointees, presidents of the regional Federal Reserve banks, and members of the financial system. Together, they decide monetary policy through information gathered both by the Board itself and by the regional banks. When people talk about the "Fed", they're usually referring to the Federal Reserve, not the federal government.
One final point: even though money from QE doesn't go to the federal government, the Federal Reserve does send money to the US Treasury on an annual basis. By law, the Federal Reserve sends its profits to the Treasury after funding its own expenses. As the linked press release states, this amounted to approximately $89 billion in January 2013.
Since the vast majority of these profits come from the Fed's bond portfolio, which in turn has grown as a result of QE, money from QE does, indirectly, flow to the US government, but not in the way you might think.
Having funds available isn't the problem, they cannot spend money that hasn't been authorized by Congress under the Antideficiency Act
Congress's authority on this matter derives from the Constitution.
Article I, Section 9, Clause 7 imposes accountability on Congressional spending:
No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of the Receipts and Expenditures of all public Money shall be published from time to time.
The first half of this clause indicates that Congress must have appropriated by law the funds to be spent before the funds can be released from the Treasury. It serves as a powerful check of the legislature on the executive branch, as it further secures Congress's power of the purse.