Questions: Fiscal Sustainability and Long-Run Debt Dynamics

Question 1

A country has debt equal to 100% of GDP, a real interest rate of 4%, and real GDP growth of 1%. What primary balance (as % of GDP) is needed to stabilize the debt ratio?

• A. A primary deficit of 3% of GDP — since the economy is growing, some deficit is sustainable
• B. A balanced primary budget — revenues must equal non-interest spending to prevent debt from rising
• C. A primary surplus of 3% of GDP — to offset the automatic snowball effect from r − g = 3%
• D. A primary surplus of 4% of GDP — to cover the full interest cost on existing debt

Answer: (C)

The debt dynamics equation: d_{t+1} = (1 + r − g)·d_t + primary deficit ratio. To stabilize d (set d_{t+1} = d_t), the primary deficit must equal −(r − g)·d_t, meaning a primary SURPLUS of (r − g)·d = 3% × 100% = 3% of GDP. Option D confuses the total interest cost (r·d = 4%) with the required primary surplus, which only needs to offset the *net* snowball (r−g). Option A is incorrect: when r > g, deficits compound the debt ratio rather than being absorbed by growth.

Question 2

When is running a primary deficit (revenues less than non-interest spending) consistent with long-run fiscal sustainability?

• A. Never — any primary deficit increases the debt stock and is therefore unsustainable
• B. Only when the debt-to-GDP ratio is below 60%, the commonly accepted safe threshold
• C. When the economy's real growth rate exceeds the real interest rate on government debt, so growth erodes the debt ratio even with modest primary deficits
• D. Only in recessions, when automatic stabilizers temporarily suppress tax revenue

Answer: (C)

When g > r, the term (1 + r − g) < 1, meaning existing debt shrinks as a share of GDP automatically — the economy grows faster than interest accumulates. In this environment, the government can run a primary deficit up to (g − r)·d without increasing the debt ratio. This is not merely a recession exception; it reflects a structurally low-interest, high-growth environment. Japan, the U.S., and eurozone countries operated with g > r for extended periods, sustaining large debts. The 60% threshold (option B) is a policy convention, not a derived sustainability criterion.

Question 3

Fiscal sustainability requires that a government eventually fully repay its outstanding debt.

• T. True
• F. False

Answer: False

Fiscal sustainability requires only that the debt-to-GDP ratio stabilize at some finite level, not that the nominal debt stock be repaid. The intertemporal government budget constraint says the present value of all future primary surpluses must equal (not exceed) the current debt. Since the economy grows over time, a constant or slowly growing debt stock falls as a share of GDP even without repayment. Governments like the UK have held debt continuously since the 18th century without 'repaying' it — what matters is that the ratio remains manageable relative to the economy's capacity to generate tax revenue.

Question 4

If a government's real interest rate suddenly rises from below to above the GDP growth rate, it must immediately run a primary surplus to prevent debt from becoming unsustainable.

• T. True
• F. False

Answer: True

This is precisely what the debt dynamics equation shows. When r crosses above g, the snowball term (1 + r − g) exceeds 1, meaning the debt ratio grows automatically each period — even with a balanced primary budget. To prevent the ratio from rising, the government must generate a primary surplus at least equal to (r − g)·d. The larger the existing debt stock d and the larger the r − g gap, the bigger the required surplus. This is the fiscal tightening mechanism behind sovereign debt crises: a rise in borrowing costs (increasing r) can flip a previously sustainable path to unsustainable almost overnight.

Question 5

Explain why the relationship between the real interest rate (r) and GDP growth rate (g) is the pivotal variable in fiscal sustainability analysis, rather than the absolute level of debt.

Model answer: The debt dynamics equation shows that the debt-to-GDP ratio evolves as d_{t+1} = (1 + r − g)·d_t + primary deficit ratio. The sign and magnitude of (r − g) determines whether debt is self-correcting or self-compounding. When g > r, each period the economy grows faster than interest accumulates, so even a constant nominal debt stock shrinks as a share of GDP — growth is the debt's natural eroder. When r > g, interest compounds faster than growth, creating a snowball: the debt ratio rises automatically, requiring an ever-increasing primary surplus just to hold it level. A country with 200% debt/GDP can be sustainable if r − g is sufficiently negative; a country with 30% debt/GDP can be unsustainable if r − g is large and positive.

Absolute debt level is misleading because what matters is the government's capacity to service that debt, which scales with GDP (the tax base). The r − g differential captures the dynamic tension between debt's growth rate and the economy's growth rate. Policymakers, markets, and the IMF all focus on this differential when assessing debt sustainability precisely because it determines the direction of the feedback loop — stabilizing or explosive.