When is debt too much? The honest answer is 'it depends', but the dependence is governed by a precise and powerful piece of arithmetic that every analyst of public finance must be able to deploy. This module is that arithmetic: the debt-dynamics equation, the all-important interest-growth differential, and why the comfortable thresholds people quote are far less solid than they sound.
The debt-dynamics equation
How the debt ratio evolves
The change in the debt-to-GDP ratio from one year to the next is governed by: Δb = (r − g) · b − pb • b = debt as a share of GDP • r = the (real) effective interest rate on the debt • g = the (real) growth rate of GDP • pb = the primary balance as a share of GDP (revenue minus NON-interest spending; positive = surplus) Reading it: the debt ratio rises by the interest-growth differential (r − g) times the existing debt — the 'snowball' of interest compounding against growth — and falls by the primary surplus. To stabilise the debt ratio (Δb = 0), the government must run a primary surplus equal to (r − g)·b. This single equation is the engine of every debt-sustainability analysis: plug in r, g, and the debt level, and it tells you the primary balance needed to keep debt from rising.
The interest-growth differential
Why r vs g changes everything
The sign of (r − g) is decisive: • If r > g (interest rate above growth) — the debt 'snowballs': even with a balanced primary budget, the debt ratio rises, because interest compounds faster than the economy grows. The government must run primary SURPLUSES just to stand still, and larger ones to reduce debt. This is the dangerous regime, and the one most African sovereigns face (high borrowing costs, especially on external commercial debt, against moderate growth). • If r < g (growth above the interest rate) — the debt ratio falls over time even with modest primary DEFICITS, because the economy outgrows the interest bill. Olivier Blanchard's 2019 presidential address ('Public Debt and Low Interest Rates') stressed that this benign regime prevailed for advanced economies for years, making debt less costly than feared. BUT — the crucial caveat for developing economies — r < g is not guaranteed and can reverse suddenly: a shock that raises r (a sudden stop, a downgrade, a currency fall raising the cost of foreign debt) or lowers g (a recession) flips the regime, and the snowball begins. Relying on r < g is a bet that can be lost overnight.
Debt-sustainability analysis
The formal application is the Debt Sustainability Analysis (DSA), the framework the IMF and others use to judge whether a country's debt is sustainable. It projects the debt path forward under baseline assumptions for r, g, and the primary balance, and then stress-tests it against shocks (a growth shock, an exchange-rate shock, a contingent-liability shock). A sustainable path is one where debt stabilises or declines and the required primary balances are politically and economically feasible. Modern DSAs use stochastic methods and fan charts to show the distribution of possible debt paths, not just a single line — an honest acknowledgement that the future is uncertain and the debt path is a probability, not a forecast.
Why thresholds are fragile
60% is not a magic line
People quote debt thresholds — 60% of GDP (the EU's Maastricht number), 70% for emerging markets, 55% for low-income countries — as if crossing them triggers crisis. They are far more fragile than that, for several reasons. First, the safe level depends entirely on the country's r, g, and ability to run primary surpluses — a country with low borrowing costs, strong growth, and a deep domestic market can sustain far more debt than one without. Second, composition matters more than the headline level: external vs domestic, short vs long maturity, foreign- vs local-currency, fixed vs floating — a country with 50% debt that is short-term and in foreign currency is far more vulnerable than one with 70% that is long-term and domestic. Third, contingent liabilities (SOEs, guarantees, the Budgeting course) hide outside the headline number. Fourth, market sentiment can flip non-linearly — debt is sustainable until suddenly it isn't, when markets refuse to roll it over (the self-fulfilling-crisis logic of module 6). So treat any threshold as a rough flag, not a bright line: the arithmetic (r, g, pb, and the composition behind them), not the headline ratio, tells you whether a debt is sustainable. The Reinhart-Rogoff '90% threshold' controversy (a famous result undermined by a spreadsheet error) is a cautionary tale against fetishising any single number.
Exercise
A country has public debt of 70% of GDP, an average real interest rate on its debt of 6%, real GDP growth of 4%, and is currently running a primary deficit of 1% of GDP. (1) Use the debt-dynamics equation to determine whether the debt ratio is rising or falling and by how much. (2) Compute the primary balance needed to stabilise the debt ratio. (3) The finance minister says 'we're fine, we're below the 80% danger threshold'. Critique this, citing at least three reasons thresholds mislead. (4) Half the debt is short-term foreign-currency Eurobonds; explain how a currency depreciation would affect the arithmetic.