Debt sustainability comes down to a race between two numbers: r, the interest rate on the debt, and g, the growth rate of the economy. If r is below g, a government can run a modest deficit before interest and the debt ratio can still fall, because the economy outgrows the debt. If r is above g, the government must run a surplus before interest just to keep the ratio from rising.
Why it matters: Whether growth or interest is winning decides if debt shrinks or snowballs.
Summary: Debt sustainability turns on the debt-dynamics identity dt = dt−1 × (1 + r) ÷ (1 + g) + pdt. When r is below g a modest primary deficit can coexist with a falling debt ratio; when r is above g the government needs a primary surplus just to hold the ratio steady.
- At 80% debt, r = 3% and g = 5%, a 0.5% primary deficit takes the ratio to 78.98% in a year; at r = 4% and g = 2% it rises to 82.07%.
- The debt-stabilizing primary balance is pb* = d × (r − g) ÷ (1 + g): 1.57% of GDP in the bad case.
- Over ten years the two cases end 32.0 points of GDP apart with identical fiscal policy.
- Average r lags market rates: the US average marketable rate rose from 1.424% (January 2022) to 3.475% (August 2026).
- Blanchard argued low r makes debt cheaper; Mauro and Zhou found marginal costs spike before defaults even when average r − g looks benign.
The single most important relationship in assessing whether a government’s debt trajectory is sustainable is the comparison between r (the average interest rate the government pays on its debt) and g (the country’s nominal GDP growth rate). When g exceeds r, a government’s debt-to-GDP ratio can actually fall over time even while running a primary deficit (a deficit before interest payments), because the economy is growing faster than the debt burden itself. When r exceeds g, the reverse dynamic takes hold — debt-to-GDP tends to rise over time unless the government runs an offsetting primary surplus large enough to compensate.
The arithmetic behind r versus g is the debt-dynamics identity. With d the debt-to-GDP ratio, r the average nominal interest rate on the debt, g nominal GDP growth and pd the primary deficit as a share of GDP:
dt = dt−1 × (1 + r) ÷ (1 + g) + pdt
Interest makes the debt grow by (1 + r); growth makes the economy that carries it grow by (1 + g); the primary deficit adds new borrowing on top. Take a country with debt at 80% of GDP, r = 3%, g = 5% and a primary deficit of 0.5% of GDP, a 2-percentage-point favorable gap. Year one: 80 × 1.03 ÷ 1.05 + 0.5 = 78.48 + 0.5 = 78.98% of GDP. The ratio falls about one point although the government borrowed more, because interest added 80 × 3% = 2.4 points while growth in the denominator removed 3.9. If growth instead slows to 2% while the interest rate rises to 4%, the same 0.5% primary deficit gives 80 × 1.04 ÷ 1.02 + 0.5 = 81.57 + 0.5 = 82.07% of GDP, and the ratio climbs every year after that.
Rearranging the identity gives the debt-stabilizing primary balance, the primary surplus that holds the ratio constant: pb* = d × (r − g) ÷ (1 + g). In the bad case, pb* = 80 × (4% − 2%) ÷ 1.02 = 1.57% of GDP: the government must swing from a 0.5% primary deficit to a 1.57% primary surplus, a tightening of about 2.1% of GDP, just to stop the ratio rising.
| Scenario (pd = 0.5% of GDP throughout) | After 1 year | After 5 years | After 10 years | Primary balance that holds debt at 80% |
|---|---|---|---|---|
| A: r = 3%, g = 5% (r − g = −2) | 78.98% | 75.07% | 70.60% | 1.52% deficit |
| B: r = 4%, g = 4% (r − g = 0) | 80.50% | 82.50% | 85.00% | Zero (balanced primary budget) |
| C: r = 4%, g = 2% (r − g = +2) | 82.07% | 90.76% | 102.61% | 1.57% surplus |
After ten years the same fiscal policy leaves debt 102.61 − 70.60 = 32.0 points of GDP apart, purely because of r and g. The flip point: with g at 5% and a 0.5% primary deficit, debt is stable when 80 × (1 + r) ÷ 1.05 + 0.5 = 80, so 1 + r = 79.5 × 1.05 ÷ 80 and r = 4.34%. A favorable gap of less than about 0.66 percentage points is not enough to carry even this small deficit. In Scenario A the ratio does not fall to zero; it heads toward the level where the deficit and the gap balance, pd × (1 + g) ÷ (g − r) = 0.5 × 1.05 ÷ 0.02 = 26.25% of GDP.
This single r-versus-g relationship is precisely why central bank interest rate decisions (Volume I’s Part 2) have such enormous, direct fiscal consequences well beyond their immediate effect on inflation and growth — a sustained period of rates rising faster than growth can flip a stable, sustainable debt trajectory into a steadily worsening one, entirely independent of any change in the government’s own spending or tax policy.
The r in the identity is the average rate on all outstanding debt, not today’s market yield, so it moves slowly: each maturing bond is refinanced at the new rate, and the speed depends on the maturity profile (Section 3.2: The Sovereign Debt Issuance Process). The average interest rate on marketable US Treasury debt was 1.424% in January 2022 and 3.475% in August 2026, more than doubling as low-coupon debt rolled over into higher yields. A government that looks at today’s r − g sees the past; the debt that matures over the next few years shows where r is heading.
In his 2019 presidential address to the American Economic Association, “Public Debt and Low Interest Rates,” Olivier Blanchard showed that since 1950 the US one-year rate had been below nominal growth in every decade except the 1980s, and the ten-year rate in four of seven decades. If that persists, he argued, debt rollovers “may well be feasible” and “public debt may have no fiscal cost”: a government can let old debt be refinanced and still see the ratio fall. He did not argue that debt has no cost at all; it can still crowd out private capital.
Paolo Mauro and Jing Zhou (IMF Working Paper 20/52, March 2020) looked at 55 countries over 200 years and reached a cooler answer to the question in their title, “Can We Sleep More Soundly?” Negative r − g is common and can last a long time, but the average r − g in the years before sovereign defaults was not higher than in normal times. What spiked was the marginal cost of new borrowing, just before default. A favorable average gap can therefore coexist with a crisis, because markets reprice new debt before the average catches up.
The evidence supports both points: r − g is the right variable for the slow arithmetic, and a poor guide to sudden loss of market access. Practitioners use both: the identity for the trend, and gross financing needs and marginal yields (Section 3.3: The Debt-to-GDP Ratio — What It Actually Measures (and Doesn’t)) for the risk of a sudden stop.
Compute the debt-stabilizing primary balance, pb* = d × (r − g) ÷ (1 + g), and compare it with the government’s actual primary balance. If the actual balance is weaker than pb* by more than about 1% of GDP, the debt ratio is on a rising path; treat any claim of sustainability as resting on a future policy change, and ask what it is. If r is below g, use the expected r over the next five years (the rate at which maturing debt will be refinanced), not the current average, because the average lags. Ignore a favorable r − g as a defense when gross financing needs are high or the debt is in foreign currency; there the risk is losing market access, which the identity does not capture.
Projecting today’s favorable r − g forward for a decade. In the table above, a government that assumes Scenario A (r = 3%, g = 5%) when Scenario C (r = 4%, g = 2%) arrives is off by 32.0 points of GDP after ten years with no change in its own policy. On GDP of $1 trillion that is about $320 billion of debt it did not plan for, and stopping the rise under Scenario C needs a primary surplus of 1.57% of GDP at the starting 80% debt, and about 2.0% once debt reaches 102.6%, instead of the 0.5% primary deficit it was running: a swing of about 2.1 to 2.5% of GDP. Avoid it by running the identity under at least one scenario in which r rises above g, and by budgeting for the debt-stabilizing balance under that scenario.
What does r minus g mean?
It is the gap between the average interest rate a government pays on its debt (r) and the nominal growth rate of the economy (g). When r − g is negative, debt grows more slowly than the economy before counting new borrowing, so the debt ratio tends to fall. When it is positive, the government needs a primary surplus just to hold the ratio steady.
What is a primary deficit?
It is the budget deficit excluding interest payments. The US deficit in fiscal 2025 was $1,775.4 billion and net interest $970.4 billion, so the primary deficit was $1,775.4B − $970.4B = $805.0 billion. The primary balance is the part of the budget a government controls this year; interest is the cost of past borrowing.
How do you calculate whether government debt is sustainable?
Use the identity dt = dt−1 × (1 + r) ÷ (1 + g) + pdt to project the debt ratio under several paths for interest rates, growth and the primary balance, then ask whether the primary balance needed to stabilize debt is politically achievable. The IMF’s debt sustainability frameworks add stress tests and gross financing needs to that core.
Can inflation reduce government debt?
Yes, temporarily. Inflation raises nominal g at once, but r on existing fixed-rate debt stays put until the debt is refinanced, so the ratio falls. Once investors demand higher yields on new debt, r catches up, and the effect fades. The longer the average maturity, the longer the window.
The debt-dynamics identity dt = dt−1 × (1 + r) ÷ (1 + g) + pdt shows why the gap between the interest rate and nominal growth drives the debt ratio: on the same 0.5% primary deficit, an 80% ratio ends ten years later at 70.60% when r is 2 points below g and at 102.61% when it is 2 points above. A favorable average r − g describes the trend but not the risk of losing market access, which shows up first in marginal borrowing costs.
Six questions on this chapter. Decide on your answer first, then click “Reveal Answer.”
1. Debt is 60% of GDP, r = 4%, g = 6% and the primary deficit is 1% of GDP. Using the debt-dynamics identity, what is debt-to-GDP after one year?
- About 61.0%
- About 59.9%
- About 58.9%
- About 62.3%
Reveal Answer
Answer: B. 60 × 1.04 ÷ 1.06 + 1 = 58.87 + 1 = 59.87%.
2. Debt is 90% of GDP, r = 5% and g = 3%. What primary balance holds the debt ratio constant?
- A primary surplus of about 1.75% of GDP
- A primary deficit of about 1.75% of GDP
- A balanced primary budget of 0% of GDP
- A primary surplus of about 2.00% of GDP
Reveal Answer
Answer: A. pb* = d × (r − g) ÷ (1 + g) = 90 × 2% ÷ 1.03 = 1.75% of GDP, a surplus because r exceeds g.
3. What did Mauro and Zhou (IMF, 2020) find about r − g before sovereign defaults?
- Negative r − g episodes were rare in their sample and never lasted more than a single year
- Defaults in their sample occurred only once r − g had exceeded about 5 percentage points
- Average r − g was not unusually high, but marginal borrowing costs spiked just before default
- Average r − g rose sharply for several years before nearly every default in their sample
Reveal Answer
Answer: C. They found negative r − g common and persistent, and that markets reprice new borrowing sharply just before default even when the average gap looks benign.
4. Why does the average interest rate on a government’s debt rise more slowly than market yields?
- Governments reset the coupon on existing debt each year to match the policy rate
- Central banks cap the yield on all outstanding government bonds at a fixed level
- Inflation automatically lowers the coupon paid on fixed-rate bonds already issued
- Only maturing debt is refinanced at new rates, so the average adjusts slowly
Reveal Answer
Answer: D. Fixed-rate debt keeps its coupon until maturity; the US average marketable rate rose from 1.424% to 3.475% between January 2022 and August 2026 as debt rolled over.
5. Worked problem: Debt is 100% of GDP, r = 4%, g = 2% and the government runs a 1% of GDP primary surplus. What is the debt ratio after a year?
Reveal Answer
Answer: d₁ = d × (1 + r) ÷ (1 + g) − pb = 100 × 1.04 ÷ 1.02 − 1 = 100.96%. The ratio still rises.
6. Worked problem: What primary balance stabilizes the ratio?
Reveal Answer
Answer: pb* = d × (r − g) ÷ (1 + g) = 100 × 0.02 ÷ 1.02 = 1.96% of GDP surplus.
- Mauro and Zhou, r minus g negative: Can We Sleep More Soundly? (IMF WP/20/52) — R − g before defaults
- US Treasury, Monthly Treasury Statement — FY2021 and FY2025 receipts, outlays, deficit and net interest
- US Treasury, Average Interest Rates on US Treasury Securities