You do not pay off a national debt, you outgrow it
Between 1960 and 1974 the American federal debt rose every single year, by two thirds in total. Over the same fourteen years it fell from 52.8% of GDP to 30.7%.
Two sentences, both true, about the same fourteen years.
The national debt rose every single year from 1960 to 1974, from $286 billion to $475 billion. The national debt fell from 52.8% of GDP to 30.7% over exactly those years.
Nothing was repaid. There was no surplus decade, no austerity programme, no year in which the outstanding total went down: not one of the fourteen year-on-year steps is negative. The debt grew by a factor of 1.66. The economy grew by a factor of 2.85.
The denominator did the work
A debt is not a quantity, it is a ratio waiting for its second number. On its own, "$475 billion" cannot tell you whether a government is in trouble, for the same reason that "a $400,000 mortgage" cannot tell you whether a household is: the figure that decides is the income underneath it.
For a country that income is GDP, and the arithmetic of the ratio is one line. If r is the average interest rate the debt carries and g is the growth rate of nominal GDP, then the ratio moves each year by
d′ = d · (1 + r) / (1 + g) − pb
where pb is the primary balance, the budget before interest. Stare at the fraction for a second and the whole postwar experience falls out of it. When g exceeds r, the multiplier is less than one, and the ratio shrinks on its own. It does not need a surplus. It does not need repayment. It needs the denominator to grow faster than the interest bill.
That is not a loophole, and it is not free money. It is what a ratio does.
How much of a deficit that permits
The uncomfortable part, for both of the usual arguments about debt, is that the permitted deficit is computable. Setting d′ = d gives the primary balance that holds the ratio exactly still:
pb* = d · (r − g) / (1 + g)
At the moment the US Treasury reports an average interest rate of about 3.4% on the debt. Put that against nominal growth a little above 4% and pb* comes out negative: a country in that position can run a primary deficit of roughly 0.9% of GDP forever and never move its debt ratio by a hair.
Flip the two numbers — interest above growth — and the same expression demands a surplus every year merely to stand still. Nothing else in the formula changed. One subtraction, r − g, decides which world a country is living in, and it is the sign that matters rather than the size.
The number that gets quoted, and the number that pays
There is a second denominator problem hiding inside the headline figure. Of the roughly $40 trillion outstanding today, about a fifth is intragovernmental: one part of the federal government holding the debt of another, principally the trust funds. That portion is a real obligation and it is also, in cash terms, the government owing itself.
Which number belongs in a sentence depends on the sentence. For "how much is owed to investors, and what does servicing it cost", the debt held by the public is the honest figure. For "what has been borrowed in total", the gross number is. Quoting the gross figure and then reasoning about interest payments to bondholders mixes the two, and it is the most common way the topic goes wrong before anyone has reached an argument.
The part that transfers to a household
A government is not a household, and the differences are the interesting part rather than a disclaimer. A state rolls its debt over indefinitely, issues the currency the debt is denominated in, and does not retire. You do none of those things. But the two mechanisms above are arithmetic, not sovereignty, and both of them turn up in a personal balance sheet.
The threshold. The tool's headline result — below a certain payment the debt is never repaid — is not a curiosity of national finance. It is the credit card. Five thousand dollars at 20% APR costs $83.33 in interest in the first month, so:
- $80 a month: the balance grows forever. You can pay for the rest of your life and owe more than you started with.
- $84 a month — four dollars over the line — clears it in 24.4 years, having paid $24,577 for a $5,000 debt.
- $100 a month: 9.0 years, $10,840.
- $200 a month: 2.7 years, $6,522.
Doubling the payment from $100 to $200 does not halve the time; it cuts it by more than two thirds, and saves four thousand dollars. That is the same pole the national numbers have, sitting a few dollars away from a minimum payment.
The ratio. The other mechanism is r − g, and your g is your pay. Take a $200,000 mortgage against a $50,000 income — a debt-to-income ratio of 4.0 — and roll it over for ten years without repaying any of it, exactly as a state rolls its own:
- Rate 3.5%, pay rising 4% a year: the ratio drifts down to 3.81. You repaid nothing and the debt got smaller relative to you.
- Rate 3.5%, pay rising 2%: it climbs to 4.63.
- Rate 6%, pay rising 3%: 5.33.
Same balance, same payments, three different outcomes, decided by a subtraction between the interest rate and your income growth. It is worth knowing which side of that subtraction you are on before deciding whether a debt is a problem.
Where the analogy stops. A country can wait: its g is an economy that grows for centuries and its debt has no due date. Your working life is a few decades and your income can stop without notice, which is why the state's escape route — outgrow it, never repay it — is available to you only slowly, only while you are earning, and not at all against a card at 20%. The arithmetic transfers. The time horizon does not.
What the fourteen years do not prove
It would be easy to read all this as reassurance, and it is not. The same arithmetic that let the ratio fall between 1960 and 1974 has been pushing it the other way for most of the time since: by 2020 it stood at 126.1%. The mechanism has no preferences. When growth outruns interest the ratio falls without anyone repaying anything, and when interest outruns growth it climbs without anyone borrowing more.
What the fourteen years do establish is narrower and worth keeping: "paying it back" is not the mechanism by which national debts have historically become smaller, and a plan that assumes it is has misread what the number is. Whether today's ratio is comfortable, and what should be done about it, are questions this page cannot settle — they turn on judgements about growth, rates and risk that are not arithmetic.
The arithmetic only tells you which quantities to argue about. That turns out to be most of the work.
References (2)
- the debt series, fetched year by year U.S. Department of the Treasury, Fiscal Data, "Historical Debt Outstanding" — annual totals from 1790.
- the denominator, 1960 onwards World Bank Open Data, NY.GDP.MKTP.CD (United States, current US$), licensed CC BY 4.0.