nice job by slacker ace

Let b be the government debt and d the primary deficit (i.e. the deficit exclusive of interest payments), both as shares of GDP. Let i be the after-tax interest rate on government borrowing and g the growth rate of GDP (both real or both nominal, it doesn't matter). Then we can rewrite the paragraph above as:

We can rearrange this to see how the debt changes from one period to the next:

Now, what happens if a given primary deficit is maintained for a long time? Does the debt-GDP ratio converge to some stable level? We can answer this question by setting the left-hand side of the above equation to zero. That gives us:

What does this mean? There are three cases to consider. If the rate of GDP growth is equal to the interest on government debt net of taxes, then the only stable primary balance is zero; any level of primary deficit leads to the debt-GDP rate rising without limit as long as its maintained. (And similarly, any level of primary surpluses leads to the government eventually paying off its debt accumulating a positive net asset position that grows without limit.) If g > i, then for any level of primary deficit, there is a corresponding stable level of debt; in this sense, there is no such thing as an "unsustainable" deficit. On the other hand, if g < i, then -- assuming debt is positive -- a constant debt requires a primary surplus.

There is a further difference between the cases. When g > i, the equilibrium is stable; if for whatever reason the debt rises or falls above the level implied by the long-run average primary deficit, it will move back toward that level over time. But when g < i, if the debt is one dollar too high, it will rise without limit; if it is one dollar too low, it will fall without limit, to be eventually replaced by an endlessly growing positive net asset position.

So, which of these three cases is most realistic? Good question! So good, in fact, I'm going to devote a whole nother post to it. The short answer: sometimes one, sometimes another. But in the US, GDP growth has exceeded pre-tax interest on 5-year Treasuries (the average maturity of US debt is around 5 years) in about 50 of the past 60 years.

The discussion up to now has been in terms of the primary balance. But nearly all public discussions of fiscal issues focus on the total deficit, which includes interest along with other categories of spending. We can rewrite the equations above in those terms, adding a superscript T to indicate we're talking about the total deficit. In these equations, g is the nominal growth rate of GDP.

Again, we define equilibrium as a situation in which the debt-GDP ratio is constant. Then we have:

In other words, any total deficit converges to a finite debt-GDP ratio. (And for every debt-GDP ratio, there is a total deficit that holds it stable.) So defining a sustainable total deficit requires picking a target debt-GDP ratio. Let's say we expect nominal GDP growth to average 5% in the future. (That's a bit low by historical standards, but it's what the CBO assumes in its long-run budget forecasts.) Then 2010's deficit of 8.8% of GDP implies a long-run debt-GDP ratio of about 175% -- a number toward the top of the range observed historically in developed countries. 175% too high? Get the long-run average deficit down to 4%, and the debt-GDP ratio converges to 80%. Deficit of 3% of GDP, debt of 60% of GDP. (Yes, the Maastricht criteria apparently assume 5% growth in nominal GDP.) It is not at all clear what the criteria are for determining the best long-run debt-GDP ratio, but that's what you've got to do before you can say whether the total deficit is too high -- or too low.

One last point: An implication of that last equation above is that if the total deficit averages zero over a long period, the debt-GDP ratio will also converge to zero. In other words, "Balance the budget over the business cycle" is another way of saying, "Pay off the whole federal debt." Yet I doubt many of the people who argue for the former, would support the latter. Which only shows how important it is to get the accounting relationships clear.