# The good, the bad and the ugly of \$22 trillion in debt

U.S. debt levels recently reached \$22 trillion, well above 100 percent of GDP.  And, obviously, debt levels will not stop there either. Is this cause for alarm? There are respectable voices on both sides of that debate.

So, what is the truth? The truth is that economists (myself included) are always of two minds, much to the chagrin of politicians. I am not here to give you a simple answer. I am here to illuminate the tradeoff, not to decide for you. I am truly sorry. You live in a democracy. That means, the choices are yours. Tough. That’s life.

The good

There is a simple bit of math to see that these debt levels might not be all that bad. In fact, it may even be possible to keep on running primary deficits and piling on more debt, and everything would be fine.

Let’s use the letter B(t) to denote the level of debt in the U.S. in year t, expressed in U.S. dollars: currently about \$22 trillion. Let’s use the letter Y(t) to denote the level of GDP in the U.S. in dollars: currently about \$20 trillion. Divide the two and you get the debt-to-GDP ratio: b(t).

Let’s also denote the primary surplus of the U.S. government by S(t): If that is negative, we have a primary deficit. How do these variables evolve? GDP grows: let’s say, at some rate g. With that, we have: Y(t+1) = (1+g) Y(t). A rate of g = 0.02, or 2-percent growth, is a good rough number for our calculation here.

Debt also grows, if no repayments are made. It grows at the rate of interest, which we denote with the letter r. That rate is currently pretty low. Let’s cautiously pick r = 0.03, or 3 percent, for this example.

Debt can be reduced by running a primary surplus, S(t). The debt in year t+1, after taking into account interest on current debt and the primary surplus S(t) is: B(t+1) = (1 + r) B(t) – S(t+1).

Divide by Y(t+1) = (1+g) Y(t) and find: b(t+1) = ((1+r)/(1+g)) b(t) – s(t+1), where s(t+1) is the ratio of the primary surplus to GDP in year t+1.

Now, if some debt ratio was alright this year, presumably that same debt ratio is alright next year, too. All we might wish to do then is keep debt ratios stable at some level where b(t+1) = b(t) = b. Plug that into the formula and use 1 + r – g to approximate ((1+r)/(1+g)) to find s(t+1) = (r – g) b.

Cool, isn’t it? If the debt-to-GDP ratio is 100 percent, then b = 1. With an interest rate of 3 percent and 2-percent growth, one just needs 1 percent of GDP as primary surplus to make sure it stays there. If the debt-to-GDP ratio is an absurdly high — 300 percent, for example — a primary surplus equal to 3 percent of GDP is good enough: still manageable. What’s there to worry about?

And there is more good news! It may well be that the interest rate is below the growth rate of the economy. In that case, the stabilization of the debt-to-GDP ratio can be done by running a small primary deficit forever. Isn’t that wonderful!?

Unfortunately, sometimes financial markets have the habit of worrying about the inability of governments to repay their debts. “But that’s never happened in the U.S.,” some will undoubtedly say. It certainly has happened in some of the U.S. states. It also happened in Europe in 2010.

It could happen at the U.S. federal level, too, or it could only be avoided via the monetary printing press, thereby ruining the worldwide trust in the U.S. dollar. That’s not a great outcome either.

Let’s use the little formula above and say that r is equal to 12 percent, a dramatically higher rate of interest.  Now, it takes a primary surplus of 10 percent of GDP to just keep things stable. Doable? Maybe. Maybe not. We would have a major fiscal crisis at hand. Bad news indeed.

Higher debt levels make for increasingly tough scenarios in case of such a destabilizing run on U.S. government debt. How much is too much? There is no precise number here. But, seriously, do we wish to find out the hard way?

The ugly