Debt sustainability

Everything about sovereign solvency falls out of one equation: whether a country grows out of its debt or drowns in it, whether austerity works or backfires, why Greece collapsed and India did not. You are going to derive that equation, not memorize it. Deriving it takes five lines and buys you a lifetime of clarity, which is why this is the single most important section here and one of the most important in all of macro-for-investors.

How is the debt dynamics equation derived?

Set up the government's budget constraint. Let D be the nominal stock of government debt and Y nominal GDP. The debt at the end of this year equals the debt we started with, plus the interest we owe on it, minus whatever primary surplus we ran to pay some of it down (or plus the primary deficit, if we ran one). Let PB be the primary balance (a surplus is positive), and i the average nominal interest rate the government pays on its debt. Then:

D_t = D_(t−1) + i·D_(t−1) − PB_t
    = D_(t−1)·(1 + i) − PB_t

Read it in words: debt grows at the interest rate, and the primary balance is the only thing pushing it back down. If you run a primary balance of exactly zero, your debt still grows at rate i forever: the interest compounds, the classic snowball.

Now convert to ratios, because the level of debt means nothing without the size of the economy carrying it. Divide through by nominal GDP Y_t. Let lower-case d = D/Y be the debt-to-GDP ratio and p = PB/Y the primary balance as a share of GDP. Nominal GDP grows at rate g (so Y_t = Y_(t−1)·(1 + g)). Then:

d_t = D_t / Y_t = [D_(t−1)·(1 + i) − PB_t] / [Y_(t−1)·(1 + g)]

d_t = [(1 + i) / (1 + g)]·d_(t−1) − p_t          ← the exact debt-dynamics equation

That boxed line is the exact law of motion for the debt ratio. Everything follows from it. Subtract d_(t−1) from both sides to get the change in the ratio:

Δd = d_t − d_(t−1) = [(1 + i)/(1 + g) − 1]·d_(t−1) − p_t
                   = [(i − g)/(1 + g)]·d_(t−1) − p_t

And for the small rates typical of a year, (i − g)/(1 + g) ≈ (i − g), and because a nominal rate minus nominal growth equals (to first order) the real rate minus real growth, the inflation in both having cancelled, we can write the famous approximate form in either nominal or real terms, as long as we are consistent:

Δd ≈ (r − g)·d − p          ← THE debt-dynamics equation

  d = debt-to-GDP ratio (a level, e.g. 0.80 for 80%)
  p = primary balance as % of GDP (surplus positive; a primary DEFICIT enters as −p, i.e. adds to Δd)
  r = the real interest rate on government debt  (or use nominal i)
  g = the real growth rate of GDP                (or use nominal g — just match r)

How do r and g drive the debt ratio?

Now extract the meaning, because this equation is the whole game. The change in a country's debt ratio has exactly two drivers:

1. The primary balance p, the policy lever. Run a primary surplus and you push d down directly, rupee for rupee. Run a primary deficit and you push it up. This is the part the government controls with today's tax-and-spend choices (which is precisely why the primary deficit page isolated the primary balance).

1. The (r − g)·d term, the "snowball," the weather the government is handed. This is the automatic drift in the debt ratio before any new borrowing, and its sign is set entirely by whether the interest rate exceeds the growth rate:

• If r < g (growth beats the interest rate): the snowball works for you. (r − g) is negative, so the term subtracts from d. The debt ratio drifts down on its own, because the economy (the denominator) is growing faster than the interest bill (the numerator) is compounding. A country in this happy regime can run primary deficits, spending more than it raises before interest, and still watch its debt/GDP fall, as long as the primary deficit is smaller than the free downward drift (g − r)·d. This is not a trick; it is how essentially every advanced economy melted its enormous WWII debts (debt/GDP well over 100% in 1945) down to modest levels by the 1970s without ever running large primary surpluses: sustained nominal growth (real growth plus a dose of inflation) simply outran the low interest rates of the era. It is also India's structural condition today. Nominal GDP growth around 10–11% comfortably exceeds the average interest rate on government debt of around 7%, so India enjoys a persistent negative snowball of roughly (0.07 − 0.11) = −4% times its debt ratio, a powerful tailwind that lets it carry ~80% general-government debt/GDP without a solvency spiral.

• If r > g (the interest rate beats growth): the snowball works against you, and that is the trap. (r − g) is positive, so the term adds to d every year, and worse, it adds more the larger d already is (it multiplies the debt stock), so the process accelerates into a true compounding trap. In this regime, merely running a balanced primary budget is not enough; the debt ratio rises anyway. To even stabilize the debt ratio you must run a primary surplus large enough to offset the snowball, and if d is large that required surplus can be brutally, politically impossibly big. This is the arithmetic of every sovereign debt crisis. It is what happened to Greece when its growth collapsed and its borrowing rate spiked simultaneously, driving (r − g) sharply positive against a 150%+ debt ratio.

What is the debt-stabilising primary balance?

The debt-stabilizing primary balance. Set Δd = 0 in the equation and solve for the primary balance that just holds the debt ratio constant:

p* = (r − g)·d          ← the primary surplus needed to STABILIZE debt/GDP

This one line is a fiscal analyst's stethoscope. Plug in a country's r, g, and d, and it tells you the primary surplus the country must run just to keep its debt ratio from rising. Compare that p* to the primary balance the country is actually running:

• If actual primary balance > p*, the debt ratio falls, and the country is solvent-and-improving.
• If actual primary balance = p*, the debt ratio holds steady.
• If actual primary balance < p*, the debt ratio rises, and if the gap persists while r > g, it rises without limit.

Two worked instances make it concrete. India: with r ≈ 7%, g ≈ 11% (nominal), d ≈ 0.83, the required primary surplus is p* = (0.07 − 0.11)·0.83 = −0.033, i.e. −3.3% of GDP. A negative required surplus means India can run a primary deficit of up to 3.3% of GDP and still see its debt ratio fall: enormous fiscal room, handed to it by g > r. A stressed Eurozone country in crisis: with r ≈ 6%, g ≈ 0% (recession), d ≈ 1.5, the required surplus is p* = (0.06 − 0.00)·1.5 = +0.09, i.e. a primary surplus of 9% of GDP just to stop the debt from growing. Almost no democracy can sustain that, which is why such countries need either a bailout, a restructuring (default), inflation, or all three.

Same equation; opposite worlds. The difference is entirely r vs g.

What are the limits of the r versus g test?

A crucial subtlety: r and g are not independent, and that is the doom loop. The naïve reading of the equation treats r, g, and p as three dials you set separately. In a crisis they are viciously linked. When a government slashes spending to raise its primary balance p (austerity), it reduces demand, which lowers g, and via the multiplier it can lower g by more than it improves p. Meanwhile, frightened bondholders demand higher yields, raising r. So an austerity program undertaken to fix the debt can drive (r − g) up faster than it improves p, and the debt ratio rises despite the austerity. This is not hypothetical. It is precisely the Greek experience of 2010–2013, where the debt ratio climbed from ~130% to ~180% during the harshest austerity in modern European history, because GDP fell about 25% (the denominator collapsed) and yields spiked (the snowball turned savage). The lesson, which the IMF itself conceded (Blanchard–Leigh, 2013, see the fiscal multiplier page), is that the multiplier and the debt equation must be solved together: when multipliers are high (a deep recession at the zero bound), front-loaded austerity can be self-defeating. Keep this coupling in mind; it is the bridge from the debt equation to the fiscal multiplier page.

One honest caveat: r < g is not a free lunch, and the debate is live. Because r < g has been the historical norm for most countries most of the time (Olivier Blanchard's 2019 AEA presidential address, "Public Debt and Low Interest Rates," made this case forcefully), some argue that debt sustainability worries are overblown for own-currency issuers. The counter, and the reason a careful analyst stays humble, is that r is not fixed: a loss of market confidence, a global rate shock, or an inflation scare can drive r above g quickly and without much warning, at which point a comfortable-looking debt ratio becomes a snowball overnight. r < g is a tailwind you are grateful for and never count on forever. We now watch both regimes play out, year by year, in a simulation.

Two debt paths, year by year: the benign regime and the trap

Nothing teaches the debt equation like running it. We take two countries that differ only in r, g, and their starting debt, hold the primary balance roughly constant, and iterate the exact law d_t = [(1 + i)/(1 + g)]·d_(t−1) − p_t for five years. Do the arithmetic yourself alongside the table. It is the muscle memory all of this exists to build.

Worked example: two debt paths, year by year

Scenario A, the benign regime (r < g), modeled on India. Start at d₀ = 80.0%. Nominal interest rate on debt i = 7%, nominal GDP growth g = 11%, and the government runs a primary deficit of 1.5% of GDP every year (so p = −1.5%; a deficit adds to debt). The coefficient is (1.07 / 1.11) = 0.96396, so each year d_t = 0.96396·d_(t−1) + 1.5.

The debt ratio falls, from 80% toward 74%, even though the government runs a deficit every single year and never once balances its budget. This is g > r doing the work: nominal growth of 11% shrinks the ratio faster than the 7% interest bill and the 1.5% primary deficit can rebuild it. Where does it settle? The steady state solves d* = −p / [1 − (1+i)/(1+g)] = 1.5 / (1 − 0.96396) = 1.5 / 0.03604 ≈ 41.6%. Left undisturbed, this country's debt drifts down over decades toward ~42% of GDP while continuously running a 1.5% primary deficit. Check it against the stabilizing-balance rule: p* = (r − g)·d = (0.07 − 0.11)·0.80 = −3.2%, meaning the country could run a primary deficit as large as 3.2% of GDP and still stabilize; running only 1.5% leaves room to spare, so debt falls.

This is the deep source of India's fiscal resilience. Not virtue, but arithmetic: a fast-growing economy is handed a powerful negative snowball, and even chronic deficits are absorbed by growth.

Scenario B, the trap (r > g), modeled on a stressed high-debt economy (think Greece 2010, or any Eurozone periphery in recession). Start at d₀ = 120.0%. Nominal interest rate i = 8% (yields have risen on fear), nominal GDP growth g = 3% (near-recession), and the government runs a smaller primary deficit of just 1.0% of GDP (p = −1.0%). The coefficient is now (1.08 / 1.03) = 1.04854, greater than one, so each year d_t = 1.04854·d_(t−1) + 1.0.

The debt ratio explodes, from 120% to 158% in five years, and the annual increase gets bigger every year (6.8 → 8.3), because the snowball multiplies an ever-larger debt stock. This country is running a smaller primary deficit than benign India (1.0% vs 1.5%), yet its debt races upward while India's melts down. The entire difference is r vs g. Note the cruelty of the stabilizing-balance test here: p* = (r − g)·d = (0.08 − 0.03)·1.20 = +6.0%. This country would need a primary surplus of 6% of GDP just to stop the debt from rising, and even a perfectly balanced primary budget (p = 0) would still see debt climb about 5.8 percentage points in year one. A 6% primary surplus means taxing far above spending, for years, during a recession: the kind of austerity that collapses g further and pushes r higher, feeding the very snowball it was meant to stop. There is often no arithmetic escape from this box without a bailout, a restructuring (default), inflation, or exiting the currency regime that trapped you, which is the menu that the sovereign debt crisis page walks through, country by country.

Put the two tables side by side and you have the whole of sovereign solvency on one page. Same equation, same kind of government running a modest primary deficit, yet one economy grows out of its debt effortlessly and the other drowns. The only material difference is the sign of (r − g). Burn p* = (r − g)·d into memory. It is to sovereign analysis what ROIC > WACC is to corporate analysis: the one inequality that separates the solvent from the doomed.