I've actually given this topic some thought, so I'll weigh in with my thoughts.
Let's say that instead of using monetary policy to combat inflation, we used fiscal policy.
In other words, we let unemployment get down to NAIRU and then rather than increase interest rates, we increase taxes to balance the budget to hold inflation down. If inflation keeps happening anyway, we keep raising taxes until it stabilizes.
This is probably the only way to actually pay off the debt without destroying the economy, by the way. As long as we don't accidentally overdue the surplus and set off deflation, keep real interest rates pretty low, and maintain positive real economic growth, the debt to GDP ratio will erode down to nothing quite quickly, as little as 20-30 years. It will be necessary to start cutting taxes to ensure the surplus isn't too big partway into the process. This isn't actually all that fanciful a scenario, incidentally, it's basically what happened in the US from about 1948 to about 1968, when the debt as a percentage of GDP plummeted from a peak of over 135% to a minimum of just under 40%, including the tax cut under Kennedy. There was not a uniform surplus in all of these, but there very frequently was and the deficit as a percentage of GDP never got above 2.5%. Arguably, Kennedy's tax cut was a little too much.
The progress towards reducing the debt was disrupted first by the arrival of peak US conventional oil production circa 1968 or so, necessitating a long term trade deficit and kicking off stagflation, and then by the insanity of the Reagan and Bush Tax Cuts. Without the Reagan revolution, it is quite likely that the US debt actually would have been down to functionally nothing by now, even after the Housing Bubble Crisis, had such a thing occurred.
So let's imagine a world without Reagan, or one in which the tide of the Tea Party is rolled back over the next generation so that counter-cyclical fiscal policy can be achieved and balanced budgets can be achieved again.
Let's say further we abandon monetary policy as a tool for economic regulation because there's no longer a market for Federal debt for the Fed to use, and we switch whole hog to fiscal policy. OK, we've just committed ourselves to maintaining price security at full employment by having the government tax more than it spends, which means it has to invest that surplus. Suddenly, a new and wholly incredible animal bursts into being, the National Credit, or perhaps the National Treasure, or more simply the US Sovereign Wealth Fund. It'll come to that well before the Debt is actually paid off, because a significant portion of the National Debt is held by the Social Security Trust Fund. Once the last privately held US Bond is repurchased by the Treasury or (more likely) the SS Trust Fund, the Trust Fund will have no choice but to invest in the larger market.
At first, it will probably start buying state and municipal bonds or the sovereign bonds of trusted US allies, or personal mortgage debts and student loan debts. More likely a combination of all of the above. Eventually though these wells will run dry, and the need to control inflation with a surplus will drive the Treasury into the corporate bond market, and finally the stock market itself.
Relatively quickly, the Federal Reserve Open Market Committee will have a new way to control the interest rate in lieu of the market for Federal debt, it will just operate with the market for Federal credit.
As the National Credit grows, a new problem emerges. An ever-escalating percentage of the government's revenue comes from interest income on the credit. By the time the National Credit is as large compared to GDP as the Debt is now, interest income will be on the order of one-fifth to one-seventh of total revenue. When the ratio of credit to GDP gets to 400% or 600%, we'll have reached the point where no taxes whatsoever are needed, and to keep the surplus from spiraling to infinity, you'll either have to greatly expand federal spending to absorb the interest income or start paying overall income subsidies. In fact, you'll have to have instituted negative income tax rates for lower income tax brackets long before this point if you want to maintain the progressive redistribution of income in the tax system.
Once this Bizarro World scenario is reached, things are very weird indeed. Almost every corporation that has outstanding debts owes that money to the government. The government owns significant stock in all publicly held corporations. The wealthy pay no taxes except perhaps an estate tax, and average workers both keep all their earnings and see a significant subsidy, but all of these workers owe the government money on their mortgages. The biggest problem now is that since the government owns so much of the economy, it has to run it. We've basically stumbled our way into the Soviet system by a very circuitous route, and there's every reason to expect that it would be horribly inefficient.
More to the point, we have an example of what a government rich enough to mostly forgo taxation looks like: Saudi Arabia. While it is difficult to see how a USA with a National Credit could lapse out of Democracy, but it's pretty easy to see a USA where flat taxation is implemented to the even further detriment of equality.
Moreover, as interest income becomes a larger portion of revenue, the temptation to stop controlling inflation with fiscal policy (eternal surpluses) and to switch to monetary policy (periods of higher real interest rates) is going to become overwhelming for politicians. In tandem with null or flat taxation, permanently higher real interest rates will make the rich richer and the poor poorer even faster.
The other problem with the above scenario is that not all inflation (or deflation) is caused by high overall employment. A serious shock limiting supply of a vital product (like the energy crisis of the 1970s) can cause inflationary pressure, while a technological development that increases the efficiency of the production of a product or products can cause deflationary pressure (like the industrialization of the 1870s to 1900). Likewise high population growth can cause inflationary pressure while low population growth can cause deflationary pressure (like contemporary Japan). A major financial crisis will still set your efforts at paying dawn debt or building credit back an entire generation, because deficit spending and zero interest rates are your only tool for fighting off deflation and unemployment. Since the Baby Boom would have caused inflationary pressure and it coincided with the biggest surpluses in recent American history, I'd say in a lower birth rate world of the future, a purely Fiscalist economy is going to be running deficits more often than surpluses.
In the real world of course, I'd prefer to keep both the tools of monetary AND fiscal policy available for intervention. I'd also like to be rid of the zero lower bound by abolishing physical currency so that monetary policy can work in a deflationary climate. I'd prefer to see inflation combated with fiscal policy instead of higher interest rates though.
I'd also like a pony.
Clearly the world where the government owns everything through the National Credit is undesirable, just as the hyperinflation world of infinite debt is undesirable, but given the choice between an economy with a small debt and a small credit, say about the size of the Social Security Trust Fund as a percentage of the economy, I'd prefer the latter.
Hopefully this musing has been helpful.
