Questions: Government Debt and Fiscal Sustainability

Question 1

Country A has debt/GDP = 100%, real interest rate r = 5%, and growth rate g = 2%. Country B has the same debt ratio but r = 2% and g = 5%. Which country faces a more urgent fiscal challenge, and why?

• A. Country A, because its interest-growth differential (r − g) is positive, so debt grows faster than the economy without primary surpluses
• B. Country B, because higher growth creates more public spending pressure
• C. Both face equal challenges since they have identical debt-to-GDP ratios
• D. Country A, because higher interest rates attract foreign capital and worsen trade balances

Answer: (A)

The law of motion Δb ≈ (r − g)·b + d shows that what matters is the interest-growth differential, not the debt level alone. Country A has r − g = +3%, so each year the debt ratio automatically rises by 3 percentage points of GDP from interest compounding alone — requiring a large primary surplus just to stabilize. Country B has r − g = −3%, meaning the economy grows faster than its debt, and the ratio falls even with modest primary deficits. Same debt, opposite trajectories.

Question 2

A government is running a primary surplus of 1% of GDP. Under what condition can the debt-to-GDP ratio still rise?

• A. When the government has high total spending regardless of revenue
• B. When the real interest rate exceeds the growth rate by enough that interest costs outpace the primary surplus
• C. When the central bank raises interest rates, making new borrowing more expensive
• D. When credit rating agencies downgrade the country's sovereign debt

Answer: (B)

The debt dynamics equation is Δb ≈ (r − g)·b + d. Even with a primary surplus (d < 0), the first term (r − g)·b can dominate if the interest-growth differential is large and the debt stock is high. For example, with b = 120%, r − g = 3%, the automatic debt increase is 3.6% of GDP per year; a 1% primary surplus leaves a net increase of 2.6% of GDP. Primary surpluses must exceed the interest-growth differential times the debt ratio to stabilize.

Question 3

A country with g > r can run a sustained primary deficit and still see its debt-to-GDP ratio decline over time.

• T. True
• F. False

Answer: True

When the growth rate exceeds the real interest rate, the economy outgrows its debt. The Δb equation shows that even with d > 0 (a primary deficit), if (r − g)·b is sufficiently negative, the debt ratio falls. This was approximately the situation in many advanced economies in the post-WWII decades, when strong growth and low interest rates allowed debt accumulated during the war to shrink relative to GDP without requiring fiscal austerity.

Question 4

A country with a 150% debt-to-GDP ratio is necessarily on an unsustainable fiscal path.

• T. True
• F. False

Answer: False

Sustainability depends on the interest-growth differential and primary balance trajectory, not the raw debt level. Japan has long maintained debt ratios well above 200% with low interest rates and domestic financing without a fiscal crisis. A high debt ratio combined with r < g and a commitment to even a small primary surplus can be entirely sustainable. Conversely, a 40% debt ratio with r >> g and persistent primary deficits is unsustainable. The debt level in isolation tells you little.

Question 5

Why is the interest-growth differential (r − g) more important than the absolute size of the debt-to-GDP ratio in determining fiscal sustainability?

Model answer: The differential determines whether existing debt compounds faster or slower than the economy grows. If r > g, each unit of debt generates interest costs that exceed the additional tax capacity created by growth, so the ratio spirals upward without ever-increasing primary surpluses. If g > r, GDP growth dilutes the debt burden automatically. The same debt level can be self-correcting or explosive depending entirely on this differential — which is why sustainability analysis centers on the trajectory implied by r, g, and the primary balance, not on the current stock of debt.

An analogy: a mortgage is manageable or crushing depending on whether your income grows faster or slower than your interest payments, not on the nominal dollar amount you borrowed. The intertemporal budget constraint formalizes this: the present value of future surpluses must cover current debt, and the discount rate for that calculation is (r − g). When r > g, the discount rate is positive, making future surpluses less valuable and required surpluses larger.