So if you google “impossible to pay off the US national debt” you’ll find an article that’s been circulating on many conservative blog sites that argues why the central banking + fractional reserve banking model is broken and that it is mathematically impossible for the US government to pay off its debt. 
I actually stumbled on that article a long time ago and it seemed to pose a very interesting paradox, and it’s been turning around in my head for the last year or so because I couldn’t find any good resources online that give a direct refutation or clarification of the phenomenon it describes. Finally today, after listening to an NPR Planet Money report on the debate between dollar bills and dollar coins and the key idea of seigniorage, I think I understand what’s going on. It’s not terribly difficult, but I guess no one else thought it was interesting enough to write down (or Google didn’t care to list such rebuttals as high as the original article in the search results).
The basic paradox.
The (most basic version of the) paradox, in a nutshell, is that the US government as of March 2012 has about $15.6 trillion in debt, while the money supply (let’s say M2) is roughly $10 trillion. (The exact numbers don’t really matter, just the fact that the outstanding debt is larger than the money supply.) Ergo, it is mathematically impossible for the US government to pay off its debts, because even if it were to collect all of the money in circulation, it would not be enough to pay off the debt. QED, right?
In fact one might even think that the situation is worse, since the M0 supply is only 1/10th of M2 (less than $1 trillion), and only M0 consists of real currency issued by the Federal Reserve. The rest of the money supply is either held as reserve balances at the Fed, or created by the money multiplier (because of fractional reserve banking).
The solution: the key point is that only money that is paid back to the Fed is removed from circulation. Most of US government debt is owed to private creditors, both domestic and foreign, and payments made to those creditors remain in the economy and therefore can be collected by the US government again through taxes, and so used to pay debts again. Namely, when the US gov’t pays debt to a private creditor, the money supply does not decrease.
In fact this point was noted in some versions of the paradox, but they dismiss it because they implicitly made the assumption that most US gov’t debt was owed to the Fed, and so when it is paid to the Fed that money is removed from circulation. A quick look at the Fed’s balance sheet reveals that the amount of debt owed to the Fed is much smaller (roughly $1.7 trillion as of April 19, 2012), and is easily covered say by M1.
Regarding the argument that “only M0 counts”, really what counts is M0 + MB, the amount of currency in circulation plus the amount of reserve balances held at the Fed. While MB is not really in circulation, it is money issued by the Fed at some point, and so in theory it can be obtained by the US government (say if a commercial bank’s overall deposits drop and it withdraws some of its reserve balances, and uses some of its withdrawals to pay taxes). It’s easy to see from the Fed balance sheets that it always holds that M0 + MB is larger than the US gov’t debt held by Fed.
A subtler paradox
There is in fact a subtler (but still resolvable) paradox lurking behind the basic one mentioned above. To describe the paradox we need to describe in a little more detail how money comes into circulation. The Fed “creates money out of thin air”, namely it is given the legal authority by the US government to create money out of nothing. This money is then put into circulation when the Fed uses it to buy assets. The Fed buys interest-accruing securities, most notably Treasury bonds.
Consider what happens when the Fed buys $100 worth of US Treasury bonds, at an interest rate say of 2%. This means that the US government now owes $102 to the Fed. But where does the US gov’t get the extra $2 to pay back the Fed? Since only the Fed has authority to create money, the US government has to issue more debt in order to obtain more dollars to fully pay off the original debt. But now this new debt has additional interest, and so this creates a snowball effect. Or does it?
The solution: there are two key points, one obvious, the other less so. The first is that the US government pays its debts in installments, and pays off parts of its principal and interest in small pieces. The second is that seigniorage occurs with the interest payments made by the US government to the Fed, and so this interest does not need to come out of further debt: while the Fed “uncreates” the money that constituted the principal of the original Treasury bond, the Fed does not “uncreate” the money that was paid as interest. Rather, the Fed transfers this money to the US Treasury (and I believe also to its other shareholders, among them commercial banks), where that money then re-enters circulation.
Therefore, as the US gov’t is paying off its Fed-owned bonds, the interest is transferred to the Treasury and so even though the Treasury initially owed $102 to the Fed, the $2 interest is in fact incrementally given back to the Treasury out of the interest payments, and so there is no snowball effect.
Other issues and questions
While this shows that, at least in theory, it is mathematically possible for the government to pay off its debt, there are further practical and/or ethical issues in the way the central banking model fundamentally works. Two notable ones include:
- While there exists enough money in circulation for the government to pay off its debt, it seems undesirable for the government to collect all this money. After all, the whole purpose of having money around is to provide liquidity in the economy. Therefore some amount of government debt seems permanently necessary in order for there to remain enough money in the economy to provide liquidity. It would be interesting to know how much debt is necessary to provide enough liquidity for a healthy economy.
- Seigniorage represents an implicit form of taxation. When the Fed creates dollars and buys Treasury bonds in an OMO, interest that would have been paid to private investors is now paid to the Fed. But since the Fed remits its profits to the US Treasury, this essentially means that the Treasury does not pay interest on bonds owned by the Fed, and so (in addition to depressing interest rates by buying Treasuries) having the Fed buy privately-held bonds saves the government on the interest payment that it would have paid to the private bond holders.
-  As far as I can tell the article is of unclear authorship and it is reposted without attribution across many different sites, so let me not link to any particular blog and rather suggest you go and find it yourself on Google. If someone does know the authorship of this article please let me know and I’ll cite it appropriately. ↩