Debt, interest and growth

A government can borrow every year while its debt shrinks relative to the economy. Whether that keeps happening depends on the interest rate it pays, how fast nominal GDP grows, and the deficit it runs before interest.

You will start with one year’s budget, then repeat it year after year: first with a fixed interest rate, then with a rate that rises as debt grows. In each section, the ledger, the diagram and the time path come from the same calculation. When you select a year, all three show that year.

Save a prediction, explore the model, then choose “I’m ready to explain what I found” to open the After question. Your answers stay in this tab only and are cleared when you reload or close the page.

EC306 Intermediate Macroeconomics. Section 3 uses the 2025 U.S. benchmark from Millington’s paper on r < g.

One year’s budget

Can debt rise while the debt ratio falls?

A government enters the year owing with GDP of , so its debt ratio is . It pays interest, and nominal GDP grows over the year. Debt, the interest rate and growth stay fixed in this section; only the primary deficit changes.

A constant interest rate

If the same budget repeats every year, where does the debt ratio go?

Now apply the year’s budget again and again, with the interest rate, growth and primary deficit held fixed. The left chart shows next year’s change in the debt ratio at each debt ratio. Where it crosses zero, the ratio stops changing: that is a steady state, also called a fixed point of the debt recurrence, b*. The right chart shows the path over time.

Interest rates that rise with debt

Can the same fiscal policy be sustainable from one starting debt and not from another?

Lenders may charge more as debt grows. Here the rate is r(b) = r0 + β(b − b̄): at the reference debt ratio of , and percentage points higher for each extra percentage point of debt. Growth is . Two economies share this rate schedule and the same primary deficit. Economy A starts at of GDP and economy B at .

About the calibration

The calibration is the U.S. benchmark in Matthew J. Millington, “Is the Government Budget Sustainable? r < g for Intermediate Macroeconomics,” Table 1 and Figure 6:

  • federal debt of of GDP and a federal primary deficit of of GDP, both for fiscal year 2025, from the U.S. Office of Management and Budget
  • nominal GDP growth of , the 2016–2025 mean, from the U.S. Bureau of Economic Analysis
  • a nominal rate of , the mean five-year Treasury yield for July 2026, from the Federal Reserve Board of Governors
  • a rate sensitivity β = , a representative value from the estimates surveyed in the paper’s Table 2

The deficit used in this section is a teaching value, chosen so that two steady states exist. At the paper’s , none does.

Try U.S. data

What do the two models say with an actual interest rate and growth rate?

This optional section takes a recent or historical five-year Treasury yield and nominal GDP growth and runs them through both models: with the rate fixed, as in Section 2, and with the rate rising with debt, as in Section 3. The two can disagree, and the panel explains why. You can keep the teaching deficit or use the actual federal primary deficit for fiscal year 2025, of GDP. It does not change your answers above.

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